Semifields, Planar Functions and MRD Codes

Home , Integral domain

Semiﬁelds, Planar Functions and MRD Codes

Yue Zhou LOOPS 2019 Conference, Budapest, July 9, 2019

National University of Defense Technology Outline

Semiﬁelds

Planar Functions

Maximum Rank-Distance Codes

1/35 Semiﬁelds In other words,

• (S, +) is an abelian group; •∗ is both left- and right-distributive over +; • there is a multiplicative identity;

• for each a and each nonzero b ∈ S, there exist unique x and y such that b ∗ x = a and y ∗ b = a.

If S does not necessarily have a multiplicative identity, then it is a presemiﬁeld.

Semiﬁelds

Deﬁnition A semiﬁeld( S, +, ∗) is a non-associative ring whose nonzero elements form a loop under multiplication.

2/35 •∗ is both left- and right-distributive over +; • there is a multiplicative identity;

• for each a and each nonzero b ∈ S, there exist unique x and y such that b ∗ x = a and y ∗ b = a.

If S does not necessarily have a multiplicative identity, then it is a presemiﬁeld.

Semiﬁelds

Deﬁnition A semiﬁeld( S, +, ∗) is a non-associative ring whose nonzero elements form a loop under multiplication.

In other words,

• (S, +) is an abelian group;

2/35 • there is a multiplicative identity;

• for each a and each nonzero b ∈ S, there exist unique x and y such that b ∗ x = a and y ∗ b = a.

If S does not necessarily have a multiplicative identity, then it is a presemiﬁeld.

Semiﬁelds

Deﬁnition A semiﬁeld( S, +, ∗) is a non-associative ring whose nonzero elements form a loop under multiplication.

In other words,

• (S, +) is an abelian group; •∗ is both left- and right-distributive over +;

2/35 • for each a and each nonzero b ∈ S, there exist unique x and y such that b ∗ x = a and y ∗ b = a.

If S does not necessarily have a multiplicative identity, then it is a presemiﬁeld.

Semiﬁelds

Deﬁnition A semiﬁeld( S, +, ∗) is a non-associative ring whose nonzero elements form a loop under multiplication.

In other words,

• (S, +) is an abelian group; •∗ is both left- and right-distributive over +; • there is a multiplicative identity;

2/35 If S does not necessarily have a multiplicative identity, then it is a presemiﬁeld.

Semiﬁelds

Deﬁnition A semiﬁeld( S, +, ∗) is a non-associative ring whose nonzero elements form a loop under multiplication.

In other words,

• (S, +) is an abelian group; •∗ is both left- and right-distributive over +; • there is a multiplicative identity;

• for each a and each nonzero b ∈ S, there exist unique x and y such that b ∗ x = a and y ∗ b = a.

2/35 Semiﬁelds

Deﬁnition A semiﬁeld( S, +, ∗) is a non-associative ring whose nonzero elements form a loop under multiplication.

In other words,

• (S, +) is an abelian group; •∗ is both left- and right-distributive over +; • there is a multiplicative identity;

• for each a and each nonzero b ∈ S, there exist unique x and y such that b ∗ x = a and y ∗ b = a.

If S does not necessarily have a multiplicative identity, then it is a presemiﬁeld.

2/35 •| S| is a prime power, and its additive group is elementary abelian.

m • S can be identiﬁed as Fq . • When ∗ is commutative, S is called a commutative semiﬁeld.

2 Dickson’s semiﬁeld (Fq, +, ∗) (1906) Let q be a power of an odd prime p, α a non-square in Fq and σ ∈ Aut(Fq). (a, b) + (c, d) := (a + c, b + d), (a, b) ∗ (c, d) := (ac + α(bd)σ, ad + bc).

Semiﬁelds

• In this talk, we only concern ﬁnite semiﬁelds.

3/35 m • S can be identiﬁed as Fq . • When ∗ is commutative, S is called a commutative semiﬁeld.

Semiﬁelds

• In this talk, we only concern ﬁnite semiﬁelds.

•| S| is a prime power, and its additive group is elementary abelian.

3/35 • When ∗ is commutative, S is called a commutative semiﬁeld.

Semiﬁelds

• In this talk, we only concern ﬁnite semiﬁelds.

•| S| is a prime power, and its additive group is elementary abelian.

m • S can be identiﬁed as Fq .

3/35 2 Dickson’s semiﬁeld (Fq, +, ∗) (1906) Let q be a power of an odd prime p, α a non-square in Fq and σ ∈ Aut(Fq). (a, b) + (c, d) := (a + c, b + d), (a, b) ∗ (c, d) := (ac + α(bd)σ, ad + bc).

Semiﬁelds

• In this talk, we only concern ﬁnite semiﬁelds.

•| S| is a prime power, and its additive group is elementary abelian.

m • S can be identiﬁed as Fq . • When ∗ is commutative, S is called a commutative semiﬁeld.

3/35 (a, b) + (c, d) := (a + c, b + d), (a, b) ∗ (c, d) := (ac + α(bd)σ, ad + bc).

Semiﬁelds

• In this talk, we only concern ﬁnite semiﬁelds.

•| S| is a prime power, and its additive group is elementary abelian.

m • S can be identiﬁed as Fq . • When ∗ is commutative, S is called a commutative semiﬁeld.

2 Dickson’s semiﬁeld (Fq, +, ∗) (1906) Let q be a power of an odd prime p, α a non-square in Fq and σ ∈ Aut(Fq).

3/35 Semiﬁelds

• In this talk, we only concern ﬁnite semiﬁelds.

•| S| is a prime power, and its additive group is elementary abelian.

m • S can be identiﬁed as Fq . • When ∗ is commutative, S is called a commutative semiﬁeld.

3/35 (a), (∞).

`∞ = {(a): a ∈ S ∪ {∞}}.

• for P 6= Q ∈ P, ∃ unique ` ∈ L such that P, Q ∈ `;

• for ` 6= m ∈ L, ∃ unique P ∈ P such that P ∈ `, m;

• there exists quadrangle, i.e. four points no three of which belong to one line.

Given a semiﬁeld (S, +, ∗), it coordinatizes a projective plane. Points: (x, y) ∈ S × S,

Lines: `a,b = {(x, a ∗ x + b): x ∈ S}, `c = {(c, y): y ∈ S},

Projective planes

P := (P, L ⊆ 2P ) is a projective plane, if

4/35 (a), (∞).

`∞ = {(a): a ∈ S ∪ {∞}}.

• for ` 6= m ∈ L, ∃ unique P ∈ P such that P ∈ `, m;

• there exists quadrangle, i.e. four points no three of which belong to one line.

Given a semiﬁeld (S, +, ∗), it coordinatizes a projective plane. Points: (x, y) ∈ S × S,

Lines: `a,b = {(x, a ∗ x + b): x ∈ S}, `c = {(c, y): y ∈ S},

Projective planes

P := (P, L ⊆ 2P ) is a projective plane, if

• for P 6= Q ∈ P, ∃ unique ` ∈ L such that P, Q ∈ `;

4/35 (a), (∞).

`∞ = {(a): a ∈ S ∪ {∞}}.

• there exists quadrangle, i.e. four points no three of which belong to one line.

Given a semiﬁeld (S, +, ∗), it coordinatizes a projective plane. Points: (x, y) ∈ S × S,

Lines: `a,b = {(x, a ∗ x + b): x ∈ S}, `c = {(c, y): y ∈ S},

Projective planes

P := (P, L ⊆ 2P ) is a projective plane, if

• for P 6= Q ∈ P, ∃ unique ` ∈ L such that P, Q ∈ `;

• for ` 6= m ∈ L, ∃ unique P ∈ P such that P ∈ `, m;

4/35 (a), (∞).

`∞ = {(a): a ∈ S ∪ {∞}}.

Given a semiﬁeld (S, +, ∗), it coordinatizes a projective plane. Points: (x, y) ∈ S × S,

Lines: `a,b = {(x, a ∗ x + b): x ∈ S}, `c = {(c, y): y ∈ S},

Projective planes

P := (P, L ⊆ 2P ) is a projective plane, if

• for P 6= Q ∈ P, ∃ unique ` ∈ L such that P, Q ∈ `;

• for ` 6= m ∈ L, ∃ unique P ∈ P such that P ∈ `, m;

• there exists quadrangle, i.e. four points no three of which belong to one line.

4/35 (a), (∞).

`∞ = {(a): a ∈ S ∪ {∞}}.

Points: (x, y) ∈ S × S,

Lines: `a,b = {(x, a ∗ x + b): x ∈ S}, `c = {(c, y): y ∈ S},

Projective planes

P := (P, L ⊆ 2P ) is a projective plane, if

• for P 6= Q ∈ P, ∃ unique ` ∈ L such that P, Q ∈ `;

• for ` 6= m ∈ L, ∃ unique P ∈ P such that P ∈ `, m;

• there exists quadrangle, i.e. four points no three of which belong to one line.

Given a semiﬁeld (S, +, ∗), it coordinatizes a projective plane.

4/35 (a), (∞).

`∞ = {(a): a ∈ S ∪ {∞}}.

Lines: `a,b = {(x, a ∗ x + b): x ∈ S}, `c = {(c, y): y ∈ S},

Projective planes

P := (P, L ⊆ 2P ) is a projective plane, if

• for P 6= Q ∈ P, ∃ unique ` ∈ L such that P, Q ∈ `;

• for ` 6= m ∈ L, ∃ unique P ∈ P such that P ∈ `, m;

• there exists quadrangle, i.e. four points no three of which belong to one line.

Given a semiﬁeld (S, +, ∗), it coordinatizes a projective plane. Points: (x, y) ∈ S × S,

4/35 `∞ = {(a): a ∈ S ∪ {∞}}.

(a), (∞).

`c = {(c, y): y ∈ S},

Projective planes

P := (P, L ⊆ 2P ) is a projective plane, if

• for P 6= Q ∈ P, ∃ unique ` ∈ L such that P, Q ∈ `;

• for ` 6= m ∈ L, ∃ unique P ∈ P such that P ∈ `, m;

• there exists quadrangle, i.e. four points no three of which belong to one line.

Given a semiﬁeld (S, +, ∗), it coordinatizes a projective plane. Points: (x, y) ∈ S × S,

Lines: `a,b = {(x, a ∗ x + b): x ∈ S},

4/35 `∞ = {(a): a ∈ S ∪ {∞}}.

(∞).

`c = {(c, y): y ∈ S},

Projective planes

P := (P, L ⊆ 2P ) is a projective plane, if

• for P 6= Q ∈ P, ∃ unique ` ∈ L such that P, Q ∈ `;

• for ` 6= m ∈ L, ∃ unique P ∈ P such that P ∈ `, m;

• there exists quadrangle, i.e. four points no three of which belong to one line.

Given a semiﬁeld (S, +, ∗), it coordinatizes a projective plane. Points: (x, y) ∈ S × S, (a),

Lines: `a,b = {(x, a ∗ x + b): x ∈ S},

4/35 (∞).

`∞ = {(a): a ∈ S ∪ {∞}}.

Projective planes

P := (P, L ⊆ 2P ) is a projective plane, if

• for P 6= Q ∈ P, ∃ unique ` ∈ L such that P, Q ∈ `;

• for ` 6= m ∈ L, ∃ unique P ∈ P such that P ∈ `, m;

• there exists quadrangle, i.e. four points no three of which belong to one line.

Given a semiﬁeld (S, +, ∗), it coordinatizes a projective plane. Points: (x, y) ∈ S × S, (a),

Lines: `a,b = {(x, a ∗ x + b): x ∈ S}, `c = {(c, y): y ∈ S},

4/35 `∞ = {(a): a ∈ S ∪ {∞}}.

Projective planes

P := (P, L ⊆ 2P ) is a projective plane, if

• for P 6= Q ∈ P, ∃ unique ` ∈ L such that P, Q ∈ `;

• for ` 6= m ∈ L, ∃ unique P ∈ P such that P ∈ `, m;

• there exists quadrangle, i.e. four points no three of which belong to one line.

Given a semiﬁeld (S, +, ∗), it coordinatizes a projective plane. Points: (x, y) ∈ S × S, (a), (∞).

Lines: `a,b = {(x, a ∗ x + b): x ∈ S}, `c = {(c, y): y ∈ S},

4/35 Projective planes

P := (P, L ⊆ 2P ) is a projective plane, if

• for P 6= Q ∈ P, ∃ unique ` ∈ L such that P, Q ∈ `;

• for ` 6= m ∈ L, ∃ unique P ∈ P such that P ∈ `, m;

• there exists quadrangle, i.e. four points no three of which belong to one line.

Given a semiﬁeld (S, +, ∗), it coordinatizes a projective plane. Points: (x, y) ∈ S × S, (a), (∞).

Lines: `a,b = {(x, a ∗ x + b): x ∈ S}, `c = {(c, y): y ∈ S}, `∞ = {(a): a ∈ S ∪ {∞}}.

4/35 If there exist m m three bijective additive mappings L, M, N : Fq → Fq such that

M(x) ? N(y) = L(x ∗ y)

m for any x, y ∈ Fq , then they are isotopic.

Theorem (Albert 1960) Two (pre)semiﬁelds coordinatize isomorphic projective planes if and only if they are isotopic.

Isotopism

Deﬁnition m m Given two (pre)semiﬁelds (Fq , +, ∗) and (Fq , +,?).

5/35 Theorem (Albert 1960) Two (pre)semiﬁelds coordinatize isomorphic projective planes if and only if they are isotopic.

Isotopism

Deﬁnition m m Given two (pre)semiﬁelds (Fq , +, ∗) and (Fq , +,?). If there exist m m three bijective additive mappings L, M, N : Fq → Fq such that

M(x) ? N(y) = L(x ∗ y)

m for any x, y ∈ Fq , then they are isotopic.

5/35 Isotopism

Deﬁnition m m Given two (pre)semiﬁelds (Fq , +, ∗) and (Fq , +,?). If there exist m m three bijective additive mappings L, M, N : Fq → Fq such that

M(x) ? N(y) = L(x ∗ y)

m for any x, y ∈ Fq , then they are isotopic.

Theorem (Albert 1960) Two (pre)semiﬁelds coordinatize isomorphic projective planes if and only if they are isotopic.

5/35 • Left nucleus: Nl (S) = { a ∈ S :(a ∗ x) ∗ y = a ∗ (x ∗ y) for all x, y ∈ S }. • Middle nucleus: Nm(S) = { a ∈ S :(x ∗ a) ∗ y = x ∗ (a ∗ y) for all x, y ∈ S }. • Right nucleus: Nr (S) = { a ∈ S :(x ∗ y) ∗ a = x ∗ (y ∗ a) for all x, y ∈ S }. • All these nuclei of semifields are invariant under isotopism and they are all finite fields.

Isotopism

• Presemiﬁelds can be “normalized” to semiﬁelds via isotopism. Given (S, +, ∗) and a 6= 0, a ∗ a is the identity of ◦ (x ∗ a) ◦ (a ∗ y) = x ∗ y.

6/35 • Middle nucleus: Nm(S) = { a ∈ S :(x ∗ a) ∗ y = x ∗ (a ∗ y) for all x, y ∈ S }. • Right nucleus: Nr (S) = { a ∈ S :(x ∗ y) ∗ a = x ∗ (y ∗ a) for all x, y ∈ S }. • All these nuclei of semifields are invariant under isotopism and they are all finite fields.

Isotopism

• Presemiﬁelds can be “normalized” to semiﬁelds via isotopism. Given (S, +, ∗) and a 6= 0, a ∗ a is the identity of ◦ (x ∗ a) ◦ (a ∗ y) = x ∗ y.

• Left nucleus: Nl (S) = { a ∈ S :(a ∗ x) ∗ y = a ∗ (x ∗ y) for all x, y ∈ S }.

6/35 • Right nucleus: Nr (S) = { a ∈ S :(x ∗ y) ∗ a = x ∗ (y ∗ a) for all x, y ∈ S }. • All these nuclei of semifields are invariant under isotopism and they are all finite fields.

Isotopism

• Presemiﬁelds can be “normalized” to semiﬁelds via isotopism. Given (S, +, ∗) and a 6= 0, a ∗ a is the identity of ◦ (x ∗ a) ◦ (a ∗ y) = x ∗ y.

• Left nucleus: Nl (S) = { a ∈ S :(a ∗ x) ∗ y = a ∗ (x ∗ y) for all x, y ∈ S }. • Middle nucleus: Nm(S) = { a ∈ S :(x ∗ a) ∗ y = x ∗ (a ∗ y) for all x, y ∈ S }.

6/35 • All these nuclei of semifields are invariant under isotopism and they are all finite fields.

Isotopism

• Presemiﬁelds can be “normalized” to semiﬁelds via isotopism. Given (S, +, ∗) and a 6= 0, a ∗ a is the identity of ◦ (x ∗ a) ◦ (a ∗ y) = x ∗ y.

• Left nucleus: Nl (S) = { a ∈ S :(a ∗ x) ∗ y = a ∗ (x ∗ y) for all x, y ∈ S }. • Middle nucleus: Nm(S) = { a ∈ S :(x ∗ a) ∗ y = x ∗ (a ∗ y) for all x, y ∈ S }. • Right nucleus: Nr (S) = { a ∈ S :(x ∗ y) ∗ a = x ∗ (y ∗ a) for all x, y ∈ S }.

6/35 Isotopism

• Presemiﬁelds can be “normalized” to semiﬁelds via isotopism. Given (S, +, ∗) and a 6= 0, a ∗ a is the identity of ◦ (x ∗ a) ◦ (a ∗ y) = x ∗ y.

6/35 • Knuth(1965)’s semifields, containing Dickson(1906)’s semifields and Hughes-Kleinfeld(1960) semifields

• Ganley commutative semiﬁelds, 1981

• Cohen-Ganley commutative semiﬁelds, 1982

• Cyclic semiﬁelds (Jha and Johnson 1989), generalizing Sandler(1962)’s semiﬁelds

• Kantor(2003) commutative semiﬁelds generalizing Knuth(1965)’s binary semiﬁelds.

• Bierbrauer, Budaghyan-Helleseth, Coulter-Matthews-Ding-Yuan, Lunardon-Marino-Polverino-Trombetti, Zha-Kyureghyan-Wang···

A long list of known semiﬁelds

• Albert’s twisted ﬁelds, 1961

7/35 • Ganley commutative semiﬁelds, 1981

• Cohen-Ganley commutative semiﬁelds, 1982

• Cyclic semiﬁelds (Jha and Johnson 1989), generalizing Sandler(1962)’s semiﬁelds

• Kantor(2003) commutative semiﬁelds generalizing Knuth(1965)’s binary semiﬁelds.

• Bierbrauer, Budaghyan-Helleseth, Coulter-Matthews-Ding-Yuan, Lunardon-Marino-Polverino-Trombetti, Zha-Kyureghyan-Wang···

A long list of known semiﬁelds

• Albert’s twisted ﬁelds, 1961

• Knuth(1965)’s semifields, containing Dickson(1906)’s semifields and Hughes-Kleinfeld(1960) semifields

7/35 • Cohen-Ganley commutative semiﬁelds, 1982

• Cyclic semiﬁelds (Jha and Johnson 1989), generalizing Sandler(1962)’s semiﬁelds

• Kantor(2003) commutative semiﬁelds generalizing Knuth(1965)’s binary semiﬁelds.

• Bierbrauer, Budaghyan-Helleseth, Coulter-Matthews-Ding-Yuan, Lunardon-Marino-Polverino-Trombetti, Zha-Kyureghyan-Wang···

A long list of known semiﬁelds

• Albert’s twisted ﬁelds, 1961

• Knuth(1965)’s semifields, containing Dickson(1906)’s semifields and Hughes-Kleinfeld(1960) semifields

• Ganley commutative semiﬁelds, 1981

7/35 • Cyclic semiﬁelds (Jha and Johnson 1989), generalizing Sandler(1962)’s semiﬁelds

• Kantor(2003) commutative semiﬁelds generalizing Knuth(1965)’s binary semiﬁelds.

• Bierbrauer, Budaghyan-Helleseth, Coulter-Matthews-Ding-Yuan, Lunardon-Marino-Polverino-Trombetti, Zha-Kyureghyan-Wang···

A long list of known semiﬁelds

• Albert’s twisted ﬁelds, 1961

• Knuth(1965)’s semifields, containing Dickson(1906)’s semifields and Hughes-Kleinfeld(1960) semifields

• Ganley commutative semiﬁelds, 1981

• Cohen-Ganley commutative semiﬁelds, 1982

7/35 • Kantor(2003) commutative semiﬁelds generalizing Knuth(1965)’s binary semiﬁelds.

• Bierbrauer, Budaghyan-Helleseth, Coulter-Matthews-Ding-Yuan, Lunardon-Marino-Polverino-Trombetti, Zha-Kyureghyan-Wang···

A long list of known semiﬁelds

• Albert’s twisted ﬁelds, 1961

• Knuth(1965)’s semifields, containing Dickson(1906)’s semifields and Hughes-Kleinfeld(1960) semifields

• Ganley commutative semiﬁelds, 1981

• Cohen-Ganley commutative semiﬁelds, 1982

• Cyclic semiﬁelds (Jha and Johnson 1989), generalizing Sandler(1962)’s semiﬁelds

7/35 • Bierbrauer, Budaghyan-Helleseth, Coulter-Matthews-Ding-Yuan, Lunardon-Marino-Polverino-Trombetti, Zha-Kyureghyan-Wang···

A long list of known semiﬁelds

• Albert’s twisted ﬁelds, 1961

• Knuth(1965)’s semifields, containing Dickson(1906)’s semifields and Hughes-Kleinfeld(1960) semifields

• Ganley commutative semiﬁelds, 1981

• Cohen-Ganley commutative semiﬁelds, 1982

• Cyclic semiﬁelds (Jha and Johnson 1989), generalizing Sandler(1962)’s semiﬁelds

• Kantor(2003) commutative semiﬁelds generalizing Knuth(1965)’s binary semiﬁelds.

7/35 A long list of known semiﬁelds

• Albert’s twisted ﬁelds, 1961

• Knuth(1965)’s semifields, containing Dickson(1906)’s semifields and Hughes-Kleinfeld(1960) semifields

• Ganley commutative semiﬁelds, 1981

• Cohen-Ganley commutative semiﬁelds, 1982

• Cyclic semiﬁelds (Jha and Johnson 1989), generalizing Sandler(1962)’s semiﬁelds

• Kantor(2003) commutative semiﬁelds generalizing Knuth(1965)’s binary semiﬁelds.

• Bierbrauer, Budaghyan-Helleseth, Coulter-Matthews-Ding-Yuan, Lunardon-Marino-Polverino-Trombetti, Zha-Kyureghyan-Wang···

7/35 • For N even, the conjecture is true (Kantor’s commutative semifields). • For N odd, the number of known semifields is less than √ c N log2(N). • The number of known commutative semifields of order N is 2 bounded above by c(logp N) .

Conjecture (Kantor,2003) The number of pairwise non-isotopic semiﬁelds of order N is not bounded above by a polynomial in N.

8/35 • For N odd, the number of known semiﬁelds is less than √ c N log2(N). • The number of known commutative semiﬁelds of order N is 2 bounded above by c(logp N) .

Conjecture (Kantor,2003) The number of pairwise non-isotopic semiﬁelds of order N is not bounded above by a polynomial in N.

• For N even, the conjecture is true (Kantor’s commutative semiﬁelds).

8/35 • The number of known commutative semiﬁelds of order N is 2 bounded above by c(logp N) .

Conjecture (Kantor,2003) The number of pairwise non-isotopic semiﬁelds of order N is not bounded above by a polynomial in N.

• For N even, the conjecture is true (Kantor’s commutative semiﬁelds). • For N odd, the number of known semiﬁelds is less than √ c N log2(N).

8/35 Conjecture (Kantor,2003) The number of pairwise non-isotopic semiﬁelds of order N is not bounded above by a polynomial in N.

• For N even, the conjecture is true (Kantor’s commutative semifields). • For N odd, the number of known semifields is less than √ c N log2(N). • The number of known commutative semifields of order N is 2 8/35 bounded above by c(logp N) . For each semifield (Fpn , +, ∗), P pi pj x ∗ y = 1≤i,j

A linearized polynomial( q-polynomial) is in Fqn [X ] of the form

q qi a0X + a1X + ··· + ai X + ··· .

Let L(n,q)[X ] denote all linearized polynomials in Fqn [X ].

n q ∼ n ∼ n×n • L(n,q)[X ]/(X − X ) = EndFq (Fq ) = Fq . • axqi ◦ bxqj = a(bxqj )qi = abqi xqi+j = abqi xqi ◦ xqj .

Cyclic semiﬁelds

Let p be a prime.

9/35 P pi pj x ∗ y = 1≤i,j

A linearized polynomial( q-polynomial) is in Fqn [X ] of the form

q qi a0X + a1X + ··· + ai X + ··· .

Let L(n,q)[X ] denote all linearized polynomials in Fqn [X ].

n q ∼ n ∼ n×n • L(n,q)[X ]/(X − X ) = EndFq (Fq ) = Fq . • axqi ◦ bxqj = a(bxqj )qi = abqi xqi+j = abqi xqi ◦ xqj .

Cyclic semiﬁelds

Let p be a prime. For each semiﬁeld (Fpn , +, ∗),

9/35 Deﬁnition

A linearized polynomial( q-polynomial) is in Fqn [X ] of the form

q qi a0X + a1X + ··· + ai X + ··· .

Let L(n,q)[X ] denote all linearized polynomials in Fqn [X ].

n q ∼ n ∼ n×n • L(n,q)[X ]/(X − X ) = EndFq (Fq ) = Fq . • axqi ◦ bxqj = a(bxqj )qi = abqi xqi+j = abqi xqi ◦ xqj .

Cyclic semiﬁelds

Let p be a prime. For each semiﬁeld (Fpn , +, ∗), P pi pj x ∗ y = 1≤i,j

9/35 Deﬁnition

A linearized polynomial( q-polynomial) is in Fqn [X ] of the form

q qi a0X + a1X + ··· + ai X + ··· .

Let L(n,q)[X ] denote all linearized polynomials in Fqn [X ].

n q ∼ n ∼ n×n • L(n,q)[X ]/(X − X ) = EndFq (Fq ) = Fq . • axqi ◦ bxqj = a(bxqj )qi = abqi xqi+j = abqi xqi ◦ xqj .

Cyclic semiﬁelds

Let p be a prime. For each semiﬁeld (Fpn , +, ∗), P pi pj x ∗ a = 1≤i,j

9/35 n q ∼ n ∼ n×n • L(n,q)[X ]/(X − X ) = EndFq (Fq ) = Fq . • axqi ◦ bxqj = a(bxqj )qi = abqi xqi+j = abqi xqi ◦ xqj .

Cyclic semiﬁelds

Let p be a prime. For each semiﬁeld (Fpn , +, ∗), P pi pj x ∗ a = 1≤i,j

A linearized polynomial( q-polynomial) is in Fqn [X ] of the form

q qi a0X + a1X + ··· + ai X + ··· .

Let L(n,q)[X ] denote all linearized polynomials in Fqn [X ].

9/35 • axqi ◦ bxqj = a(bxqj )qi = abqi xqi+j = abqi xqi ◦ xqj .

Cyclic semiﬁelds

Let p be a prime. For each semiﬁeld (Fpn , +, ∗), P pi pj x ∗ a = 1≤i,j

A linearized polynomial( q-polynomial) is in Fqn [X ] of the form

q qi a0X + a1X + ··· + ai X + ··· .

Let L(n,q)[X ] denote all linearized polynomials in Fqn [X ].

n q ∼ n ∼ n×n • L(n,q)[X ]/(X − X ) = EndFq (Fq ) = Fq .

9/35 Cyclic semiﬁelds

Let p be a prime. For each semiﬁeld (Fpn , +, ∗), P pi pj x ∗ a = 1≤i,j

A linearized polynomial( q-polynomial) is in Fqn [X ] of the form

q qi a0X + a1X + ··· + ai X + ··· .

Let L(n,q)[X ] denote all linearized polynomials in Fqn [X ].

n q ∼ n ∼ n×n • L(n,q)[X ]/(X − X ) = EndFq (Fq ) = Fq . • axqi ◦ bxqj = a(bxqj )qi = abqi xqi+j = abqi xqi ◦ xqj .

9/35 Cyclic semiﬁelds

Let p be a prime. For each semiﬁeld (Fpn , +, ∗), P pi pj x ∗ a = 1≤i,j

A linearized polynomial( q-polynomial) is in Fqn [X ] of the form

q qi a0X + a1X + ··· + ai X + ··· .

Let L(n,q)[X ] denote all linearized polynomials in Fqn [X ].

n q ∼ n ∼ n×n • L(n,q)[X ]/(X − X ) = EndFq (Fq ) = Fq . • ax qi ◦ bx qj = a(bxqj )qi = abqi xqi+j = abqi xqi ◦ xqj .

9/35 • R is a non-commutative integral domain. • R is both left- and right-Euclidean domain, with the usual degree function. • Let f be an irreducible polynomial in R, deg(f ) = s, S = {g ∈ R : deg(g) ≤ s − 1}. Define g ∗ h := gh n mod r f (X ). • (S, +, ∗) is a semifield. q ∼ • (Fqn [X ;(·) ], +, ·) = (L(n,q)[X ], +, ◦) and take f (X ) = X − 1, q n ∼ qn (Fqn [X ;(·) ], +, ·)/(X − 1) = (L(n,q)[X ], +, ◦)/(X − X ). √ n • For given N = q , there are at most N log2(N) cyclic semifields (Kantor,Liebler 2008).

Cyclic semiﬁelds

σ qi For σ ∈ Gal(Fqn /Fq), i.e. x = x , let R = Fqn [X ; σ] be a skew polynomial ring in which Xa = aσX .

10/35 • R is both left- and right-Euclidean domain, with the usual degree function. • Let f be an irreducible polynomial in R, deg(f ) = s, S = {g ∈ R : deg(g) ≤ s − 1}. Define g ∗ h := gh n mod r f (X ). • (S, +, ∗) is a semifield. q ∼ • (Fqn [X ;(·) ], +, ·) = (L(n,q)[X ], +, ◦) and take f (X ) = X − 1, q n ∼ qn (Fqn [X ;(·) ], +, ·)/(X − 1) = (L(n,q)[X ], +, ◦)/(X − X ). √ n • For given N = q , there are at most N log2(N) cyclic semifields (Kantor,Liebler 2008).

Cyclic semiﬁelds

σ qi For σ ∈ Gal(Fqn /Fq), i.e. x = x , let R = Fqn [X ; σ] be a skew polynomial ring in which Xa = aσX . • R is a non-commutative integral domain.

10/35 • Let f be an irreducible polynomial in R, deg(f ) = s, S = {g ∈ R : deg(g) ≤ s − 1}. Define g ∗ h := gh n mod r f (X ). • (S, +, ∗) is a semifield. q ∼ • (Fqn [X ;(·) ], +, ·) = (L(n,q)[X ], +, ◦) and take f (X ) = X − 1, q n ∼ qn (Fqn [X ;(·) ], +, ·)/(X − 1) = (L(n,q)[X ], +, ◦)/(X − X ). √ n • For given N = q , there are at most N log2(N) cyclic semifields (Kantor,Liebler 2008).

Cyclic semiﬁelds

10/35 • (S, +, ∗) is a semiﬁeld. q ∼ • (Fqn [X ;(·) ], +, ·) = (L(n,q)[X ], +, ◦) and take f (X ) = X − 1, q n ∼ qn (Fqn [X ;(·) ], +, ·)/(X − 1) = (L(n,q)[X ], +, ◦)/(X − X ). √ n • For given N = q , there are at most N log2(N) cyclic semiﬁelds (Kantor,Liebler 2008).

Cyclic semiﬁelds

10/35 q ∼ • (Fqn [X ;(·) ], +, ·) = (L(n,q)[X ], +, ◦) and take f (X ) = X − 1, q n ∼ qn (Fqn [X ;(·) ], +, ·)/(X − 1) = (L(n,q)[X ], +, ◦)/(X − X ). √ n • For given N = q , there are at most N log2(N) cyclic semiﬁelds (Kantor,Liebler 2008).

Cyclic semiﬁelds

10/35 and take f (X ) = X − 1, q n ∼ qn (Fqn [X ;(·) ], +, ·)/(X − 1) = (L(n,q)[X ], +, ◦)/(X − X ). √ n • For given N = q , there are at most N log2(N) cyclic semiﬁelds (Kantor,Liebler 2008).

Cyclic semiﬁelds

σ qi For σ ∈ Gal(Fqn /Fq), i.e. x = x , let R = Fqn [X ; σ] be a skew polynomial ring in which Xa = aσX . • R is a non-commutative integral domain. • R is both left- and right-Euclidean domain, with the usual degree function. • Let f be an irreducible polynomial in R, deg(f ) = s, S = {g ∈ R : deg(g) ≤ s − 1}. Deﬁne g ∗ h := gh n mod r f (X ). • (S, +, ∗) is a semiﬁeld. q ∼ • (Fqn [X ;(·) ], +, ·) = (L(n,q)[X ], +, ◦)

10/35 √ n • For given N = q , there are at most N log2(N) cyclic semiﬁelds (Kantor,Liebler 2008).

Cyclic semiﬁelds

10/35 Cyclic semiﬁelds

semiﬁelds (Kantor,Liebler 2008). 10/35 Deﬁne x ∗ y = xy − axσy τ .

Then (Fq, +, ∗) is a presemiﬁeld, called Albert’s twisted ﬁeld.

By twisting some other known semiﬁelds, people have found more new semiﬁelds.

• Pott-Z. commutative semifields, 2013. • Dempwolff semifields, 2013. • Twisted cyclic semifields by Sheekey, arxiv.

Twisting approach

Deﬁnition σ−1 τ−1 Let σ, τ ∈ Aut(Fq) and a ∈ Fq such that a = x y has no solution.

11/35 Then (Fq, +, ∗) is a presemiﬁeld, called Albert’s twisted ﬁeld.

By twisting some other known semiﬁelds, people have found more new semiﬁelds.

• Pott-Z. commutative semifields, 2013. • Dempwolff semifields, 2013. • Twisted cyclic semifields by Sheekey, arxiv.

Twisting approach

Deﬁnition σ−1 τ−1 Let σ, τ ∈ Aut(Fq) and a ∈ Fq such that a = x y has no solution. Deﬁne x ∗ y = xy − axσy τ .

11/35 By twisting some other known semiﬁelds, people have found more new semiﬁelds.

• Pott-Z. commutative semifields, 2013. • Dempwolff semifields, 2013. • Twisted cyclic semifields by Sheekey, arxiv.

Twisting approach

Deﬁnition σ−1 τ−1 Let σ, τ ∈ Aut(Fq) and a ∈ Fq such that a = x y has no solution. Deﬁne x ∗ y = xy − axσy τ .

Then (Fq, +, ∗) is a presemiﬁeld, called Albert’s twisted ﬁeld.

11/35 • Pott-Z. commutative semifields, 2013. • Dempwolff semifields, 2013. • Twisted cyclic semifields by Sheekey, arxiv.

Twisting approach

Deﬁnition σ−1 τ−1 Let σ, τ ∈ Aut(Fq) and a ∈ Fq such that a = x y has no solution. Deﬁne x ∗ y = xy − axσy τ .

Then (Fq, +, ∗) is a presemiﬁeld, called Albert’s twisted ﬁeld.

By twisting some other known semiﬁelds, people have found more new semiﬁelds.

11/35 • Dempwolff semifields, 2013. • Twisted cyclic semifields by Sheekey, arxiv.

Twisting approach

Deﬁnition σ−1 τ−1 Let σ, τ ∈ Aut(Fq) and a ∈ Fq such that a = x y has no solution. Deﬁne x ∗ y = xy − axσy τ .

Then (Fq, +, ∗) is a presemiﬁeld, called Albert’s twisted ﬁeld.

By twisting some other known semiﬁelds, people have found more new semiﬁelds.

• Pott-Z. commutative semiﬁelds, 2013.

11/35 • Twisted cyclic semiﬁelds by Sheekey, arxiv.

Twisting approach

Deﬁnition σ−1 τ−1 Let σ, τ ∈ Aut(Fq) and a ∈ Fq such that a = x y has no solution. Deﬁne x ∗ y = xy − axσy τ .

Then (Fq, +, ∗) is a presemiﬁeld, called Albert’s twisted ﬁeld.

By twisting some other known semiﬁelds, people have found more new semiﬁelds.

• Pott-Z. commutative semifields, 2013. • Dempwolff semifields, 2013.

11/35 Twisting approach

Deﬁnition σ−1 τ−1 Let σ, τ ∈ Aut(Fq) and a ∈ Fq such that a = x y has no solution. Deﬁne x ∗ y = xy − axσy τ .

Then (Fq, +, ∗) is a presemiﬁeld, called Albert’s twisted ﬁeld.

By twisting some other known semiﬁelds, people have found more new semiﬁelds.

• Pott-Z. commutative semifields, 2013. • Dempwolff semifields, 2013. • Twisted cyclic semifields by Sheekey, arxiv.

11/35 • (Fq2 , +, ∗) is a commutative presemiﬁeld (Pott, Z. 2013). • This construction has two parameters k and σ. • Let q = pm where p is a prime, m = 2e µ with gcd(µ, 2) = 1. µ m This construction gives exactly b 2 cd 2 e non-isotopic presemiﬁelds.

2 • It gives us the bound c(logp q) of known commutative semiﬁelds.

σ σ k σ • It is isotopic to x ◦k y := x y + xy where σ = p (y 7→ y ). • (a, b) ∗ (c, d) := (ac + α(bd)σ, ad + bc) (Dickson’s semiﬁelds), where α a non-square.

Twisting approach

• By taking τ = σ−1 and a = −1, one has x ◦ y = xy + xσy σ−1 .

12/35 • (Fq2 , +, ∗) is a commutative presemiﬁeld (Pott, Z. 2013). • This construction has two parameters k and σ. • Let q = pm where p is a prime, m = 2e µ with gcd(µ, 2) = 1. µ m This construction gives exactly b 2 cd 2 e non-isotopic presemiﬁelds.

2 • It gives us the bound c(logp q) of known commutative semiﬁelds.

• (a, b) ∗ (c, d) := (ac + α(bd)σ, ad + bc) (Dickson’s semiﬁelds), where α a non-square.

Twisting approach

• By taking τ = σ−1 and a = −1, one has x ◦ y = xy + xσy σ−1 .

σ σ k σ • It is isotopic to x ◦k y := x y + xy where σ = p (y 7→ y ).

2 • It gives us the bound c(logp q) of known commutative semiﬁelds.

Twisting approach

• By taking τ = σ−1 and a = −1, one has x ◦ y = xy + xσy σ−1 .

σ σ k σ • It is isotopic to x ◦k y := x y + xy where σ = p (y 7→ y ). • (a, b) ∗ (c, d) := (ac + α(bd)σ, ad + bc) (Dickson’s semiﬁelds), where α a non-square.

2 • It gives us the bound c(logp q) of known commutative semiﬁelds.

Twisting approach

• By taking τ = σ−1 and a = −1, one has x ◦ y = xy + xσy σ−1 .

σ σ k σ • It is isotopic to x ◦k y := x y + xy where σ = p (y 7→ y ). σ • (a, b) ∗ (c, d) := (a ◦k c + α(b ◦k d) , ad + bc), where α a non-square.

12/35 • This construction has two parameters k and σ. • Let q = pm where p is a prime, m = 2e µ with gcd(µ, 2) = 1. µ m This construction gives exactly b 2 cd 2 e non-isotopic presemiﬁelds.

2 • It gives us the bound c(logp q) of known commutative semiﬁelds.

Twisting approach

• By taking τ = σ−1 and a = −1, one has x ◦ y = xy + xσy σ−1 .

σ σ k σ • It is isotopic to x ◦k y := x y + xy where σ = p (y 7→ y ). σ • (a, b) ∗ (c, d) := (a ◦k c + α(b ◦k d) , ad + bc), where α a non-square.

• (Fq2 , +, ∗) is a commutative presemiﬁeld (Pott, Z. 2013).

12/35 • Let q = pm where p is a prime, m = 2e µ with gcd(µ, 2) = 1. µ m This construction gives exactly b 2 cd 2 e non-isotopic presemiﬁelds.

2 • It gives us the bound c(logp q) of known commutative semiﬁelds.

Twisting approach

• By taking τ = σ−1 and a = −1, one has x ◦ y = xy + xσy σ−1 .

σ σ k σ • It is isotopic to x ◦k y := x y + xy where σ = p (y 7→ y ). σ • (a, b) ∗ (c, d) := (a ◦k c + α(b ◦k d) , ad + bc), where α a non-square.

• (Fq2 , +, ∗) is a commutative presemiﬁeld (Pott, Z. 2013). • This construction has two parameters k and σ.

12/35 2 • It gives us the bound c(logp q) of known commutative semiﬁelds.

Twisting approach

• By taking τ = σ−1 and a = −1, one has x ◦ y = xy + xσy σ−1 .

σ σ k σ • It is isotopic to x ◦k y := x y + xy where σ = p (y 7→ y ). σ • (a, b) ∗ (c, d) := (a ◦k c + α(b ◦k d) , ad + bc), where α a non-square.

• (Fq2 , +, ∗) is a commutative presemiﬁeld (Pott, Z. 2013). • This construction has two parameters k and σ. • Let q = pm where p is a prime, m = 2e µ with gcd(µ, 2) = 1. µ m This construction gives exactly b 2 cd 2 e non-isotopic presemiﬁelds.

12/35 Twisting approach

• By taking τ = σ−1 and a = −1, one has x ◦ y = xy + xσy σ−1 .

σ σ k σ • It is isotopic to x ◦k y := x y + xy where σ = p (y 7→ y ). σ • (a, b) ∗ (c, d) := (a ◦k c + α(b ◦k d) , ad + bc), where α a non-square.

2 • It gives us the bound c(logp q) of known commutative semiﬁelds.

12/35 • Dempwolff’s semifields (u, v) ∗ (x, y) = (ux + v ◦ y, uy + ζ(v ◦ x)) where ◦ defines a twisted field. • Let f be an irreducible polynomial in R, deg(f ) = s, P i S = {g = gi X ∈ Fqn [X ; σ] : deg(g) ≤ s − 1}. n Define g ∗ h := gh mod r f (X ) for g, h ∈ S.(S, +, ∗) is a cyclic semifield. • Hughes-Kleinfeld (1960) semifields and generalized Dickson’s (1965) semifields actually belong to this new construction. • Question: how may non-isotopic twisted cyclic semifields are there?

Twisting approach

x ◦ y = xy − axσy τ

13/35 • Hughes-Kleinfeld (1960) semifields and generalized Dickson’s (1965) semifields actually belong to this new construction. • Question: how may non-isotopic twisted cyclic semifields are there?

• Let f be an irreducible polynomial in R, deg(f ) = s, P i S = {g = gi X ∈ Fqn [X ; σ] : deg(g) ≤ s − 1}. n Deﬁne g ∗ h := gh mod r f (X ) for g, h ∈ S.(S, +, ∗) is a cyclic semiﬁeld.

Twisting approach

x ◦ y = xy − axσy τ

• Dempwolff’s semifields (u, v) ∗ (x, y) = (ux + v ◦ y, uy + ζ(v ◦ x)) where ◦ defines a twisted field.

Twisting approach

x ◦ y = xy − axσy τ

• Dempwolff’s semifields (u, v) ∗ (x, y) = (ux + v ◦ y, uy + ζ(v ◦ x)) where ◦ defines a twisted field. • Let f be an irreducible polynomial in R, deg(f ) = s, P i S = {g = gi X ∈ Fqn [X ; σ] : deg(g) ≤ s − 1}. n Define g ∗ h := gh mod r f (X ) for g, h ∈ S.(S, +, ∗) is a cyclic semifield.

Twisting approach

x ◦ c = cx − acτ xσ

Twisting approach

x ◦ c = cx − acτ xσ

13/35 • Question: how may non-isotopic twisted cyclic semiﬁelds are there?

Twisting approach

x ◦ c = cx − acτ xσ

• Dempwolff’s semifields (u, v) ∗ (x, y) = (ux + v ◦ y, uy + ζ(v ◦ x)) where ◦ defines a twisted field. • Let f be an irreducible polynomial in R, deg(f ) = s, P i ρ S = {g = gi X ∈ Fqn [X ; σ] : deg(g) ≤ s, gs = ηg0 }. n Define g ∗ h := gh mod r f (X ) for g, h ∈ S.(S, +, ∗) is a twisted cyclic semifield (Sheekey arxiv). • Hughes-Kleinfeld (1960) semifields and generalized Dickson’s (1965) semifields actually belong to this new construction.

13/35 Twisting approach

x ◦ c = cx − acτ xσ

Then (F, +, ∗) is a presemiﬁeld. • When m = 1, they are Knuth’s binary presemiﬁelds.

• For almost each diﬀerent choice of (ζ1, ··· , ζm), the presemiﬁelds are non-isotopic.

Kantor’s commutative semiﬁelds

Given a chain of ﬁelds F = F0 ⊃ F1 ⊃ · · · ⊃ Fm of characteristic 2 with [F : Fm] odd and corresponding trace mappings Tri : F → Fi , ζi ∈ F.

14/35 • When m = 1, they are Knuth’s binary presemiﬁelds.

• For almost each diﬀerent choice of (ζ1, ··· , ζm), the presemiﬁelds are non-isotopic.

Kantor’s commutative semiﬁelds

Given a chain of ﬁelds F = F0 ⊃ F1 ⊃ · · · ⊃ Fm of characteristic 2 with [F : Fm] odd and corresponding trace mappings Tri : F → Fi , ζi ∈ F.

m m !2 X X x ∗ y := xy + x Tri (ζi y) + y Tri (ζi x) . i=1 i=1

Then (F, +, ∗) is a presemiﬁeld.

14/35 • For almost each diﬀerent choice of (ζ1, ··· , ζm), the presemiﬁelds are non-isotopic.

Kantor’s commutative semiﬁelds

Given a chain of ﬁelds F = F0 ⊃ F1 ⊃ · · · ⊃ Fm of characteristic 2 with [F : Fm] odd and corresponding trace mappings Tri : F → Fi , ζi ∈ F.

m m !2 X X x ∗ y := xy + x Tri (ζi y) + y Tri (ζi x) . i=1 i=1

Then (F, +, ∗) is a presemiﬁeld. • When m = 1, they are Knuth’s binary presemiﬁelds.

14/35 Kantor’s commutative semiﬁelds

Given a chain of ﬁelds F = F0 ⊃ F1 ⊃ · · · ⊃ Fm of characteristic 2 with [F : Fm] odd and corresponding trace mappings Tri : F → Fi , ζi ∈ F.

m m !2 X X x ∗ y := xy + x Tri (ζi y) + y Tri (ζi x) . i=1 i=1

Then (F, +, ∗) is a presemiﬁeld. • When m = 1, they are Knuth’s binary presemiﬁelds.

• For almost each diﬀerent choice of (ζ1, ··· , ζm), the presemiﬁelds are non-isotopic.

14/35 Planar Functions • Deﬁne f : Fq → Fq by f (x) := x ∗ x. • For every a 6= 0, x 7→ f (x + a) − f (x) = 2a ∗ x + a ∗ a is a permutation on Fq. Deﬁnition (Dembowski and Ostrom 1968)

Let f : Fq → Fq. If for every a 6= 0, f (x + a) − f (x) deﬁnes a permutation on Fq, then f is a planar function.

• Commutative (pre)semiﬁelds (q odd) ⇒ Planar functions. • Planar functions from commutative semiﬁelds can always be written as a Dembowski-Ostrom (DO) polynomial P pi +pj aij x . σ σ k pk +1 • x ◦k y := x y + xy with σ = p , f (x) = 2x .

Planar functions (q odd)

• Take a commutative (pre)semiﬁeld (Fq, +, ∗), where q is odd.

15/35 • For every a 6= 0, x 7→ f (x + a) − f (x) = 2a ∗ x + a ∗ a is a permutation on Fq. Deﬁnition (Dembowski and Ostrom 1968)

Let f : Fq → Fq. If for every a 6= 0, f (x + a) − f (x) deﬁnes a permutation on Fq, then f is a planar function.

Planar functions (q odd)

• Take a commutative (pre)semiﬁeld (Fq, +, ∗), where q is odd.

• Deﬁne f : Fq → Fq by f (x) := x ∗ x.

15/35 Deﬁnition (Dembowski and Ostrom 1968)

Let f : Fq → Fq. If for every a 6= 0, f (x + a) − f (x) deﬁnes a permutation on Fq, then f is a planar function.

Planar functions (q odd)

• Take a commutative (pre)semiﬁeld (Fq, +, ∗), where q is odd.

• Deﬁne f : Fq → Fq by f (x) := x ∗ x. • For every a 6= 0, x 7→ f (x + a) − f (x) = 2a ∗ x + a ∗ a is a permutation on Fq.

15/35 • Commutative (pre)semiﬁelds (q odd) ⇒ Planar functions. • Planar functions from commutative semiﬁelds can always be written as a Dembowski-Ostrom (DO) polynomial P pi +pj aij x . σ σ k pk +1 • x ◦k y := x y + xy with σ = p , f (x) = 2x .

Planar functions (q odd)

• Take a commutative (pre)semiﬁeld (Fq, +, ∗), where q is odd.

• Deﬁne f : Fq → Fq by f (x) := x ∗ x. • For every a 6= 0, x 7→ f (x + a) − f (x) = 2a ∗ x + a ∗ a is a permutation on Fq. Deﬁnition (Dembowski and Ostrom 1968)

Let f : Fq → Fq. If for every a 6= 0, f (x + a) − f (x) deﬁnes a permutation on Fq, then f is a planar function.

15/35 • Planar functions from commutative semiﬁelds can always be written as a Dembowski-Ostrom (DO) polynomial P pi +pj aij x . σ σ k pk +1 • x ◦k y := x y + xy with σ = p , f (x) = 2x .

Planar functions (q odd)

• Take a commutative (pre)semiﬁeld (Fq, +, ∗), where q is odd.

• Deﬁne f : Fq → Fq by f (x) := x ∗ x. • For every a 6= 0, x 7→ f (x + a) − f (x) = 2a ∗ x + a ∗ a is a permutation on Fq. Deﬁnition (Dembowski and Ostrom 1968)

Let f : Fq → Fq. If for every a 6= 0, f (x + a) − f (x) deﬁnes a permutation on Fq, then f is a planar function.

• Commutative (pre)semiﬁelds (q odd) ⇒ Planar functions.

15/35 σ σ k pk +1 • x ◦k y := x y + xy with σ = p , f (x) = 2x .

Planar functions (q odd)

• Take a commutative (pre)semiﬁeld (Fq, +, ∗), where q is odd.

• Deﬁne f : Fq → Fq by f (x) := x ∗ x. • For every a 6= 0, x 7→ f (x + a) − f (x) = 2a ∗ x + a ∗ a is a permutation on Fq. Deﬁnition (Dembowski and Ostrom 1968)

Let f : Fq → Fq. If for every a 6= 0, f (x + a) − f (x) deﬁnes a permutation on Fq, then f is a planar function.

• Commutative (pre)semiﬁelds (q odd) ⇒ Planar functions. • Planar functions from commutative semiﬁelds can always be written as a Dembowski-Ostrom (DO) polynomial P pi +pj aij x .

15/35 Planar functions (q odd)

• Take a commutative (pre)semiﬁeld (Fq, +, ∗), where q is odd.

• Deﬁne f : Fq → Fq by f (x) := x ∗ x. • For every a 6= 0, x 7→ f (x + a) − f (x) = 2a ∗ x + a ∗ a is a permutation on Fq. Deﬁnition (Dembowski and Ostrom 1968)

Let f : Fq → Fq. If for every a 6= 0, f (x + a) − f (x) deﬁnes a permutation on Fq, then f is a planar function.

3k +1 • Counter-example: (1997) Coulter-Matthews monomials x 2 over F3n . • Other examples? p > 3?

• By a planar function f over Fq, one can always deﬁne an aﬃne plane Πf using D = {(x, f (x)) : x ∈ Fq}.

Points: (x, y) ∈ Fq × Fq

Lines: `a,b = D + (a, b) = {(x + a, f (x) + b): x ∈ Fq}, `a = {(a, y): y ∈ Fq}.

Planar functions (q odd)

• Planar functions ⇒ commutative (pre)semiﬁelds?

16/35 3k +1 • Counter-example: (1997) Coulter-Matthews monomials x 2 over F3n . • Other examples? p > 3?

• By a planar function f over Fq, one can always deﬁne an aﬃne plane Πf using D = {(x, f (x)) : x ∈ Fq}.

Points: (x, y) ∈ Fq × Fq

Lines: `a,b = D + (a, b) = {(x + a, f (x) + b): x ∈ Fq}, `a = {(a, y): y ∈ Fq}.

Planar functions (q odd)

• Planar functions ⇒ commutative (pre)semiﬁelds? (Conjectured by Dembowksi and Ostrom 1968)

16/35 • Other examples? p > 3?

• By a planar function f over Fq, one can always deﬁne an aﬃne plane Πf using D = {(x, f (x)) : x ∈ Fq}.

Points: (x, y) ∈ Fq × Fq

Lines: `a,b = D + (a, b) = {(x + a, f (x) + b): x ∈ Fq}, `a = {(a, y): y ∈ Fq}.

Planar functions (q odd)

• Planar functions ⇒ commutative (pre)semiﬁelds? (Conjectured by Dembowksi and Ostrom 1968)

3k +1 • Counter-example: (1997) Coulter-Matthews monomials x 2 over F3n .

16/35 • By a planar function f over Fq, one can always deﬁne an aﬃne plane Πf using D = {(x, f (x)) : x ∈ Fq}.

Points: (x, y) ∈ Fq × Fq

Lines: `a,b = D + (a, b) = {(x + a, f (x) + b): x ∈ Fq}, `a = {(a, y): y ∈ Fq}.

Planar functions (q odd)

• Planar functions ⇒ commutative (pre)semiﬁelds? (Conjectured by Dembowksi and Ostrom 1968)

3k +1 • Counter-example: (1997) Coulter-Matthews monomials x 2 over F3n . • Other examples? p > 3?

16/35 Points: (x, y) ∈ Fq × Fq

Lines: `a,b = D + (a, b) = {(x + a, f (x) + b): x ∈ Fq}, `a = {(a, y): y ∈ Fq}.

Planar functions (q odd)

• Planar functions ⇒ commutative (pre)semiﬁelds? (Conjectured by Dembowksi and Ostrom 1968)

3k +1 • Counter-example: (1997) Coulter-Matthews monomials x 2 over F3n . • Other examples? p > 3?

• By a planar function f over Fq, one can always deﬁne an aﬃne plane Πf using D = {(x, f (x)) : x ∈ Fq}.

16/35 Lines: `a,b = D + (a, b) = {(x + a, f (x) + b): x ∈ Fq}, `a = {(a, y): y ∈ Fq}.

Planar functions (q odd)

• Planar functions ⇒ commutative (pre)semiﬁelds? (Conjectured by Dembowksi and Ostrom 1968)

3k +1 • Counter-example: (1997) Coulter-Matthews monomials x 2 over F3n . • Other examples? p > 3?

• By a planar function f over Fq, one can always deﬁne an aﬃne plane Πf using D = {(x, f (x)) : x ∈ Fq}.

Points: (x, y) ∈ Fq × Fq

16/35 Planar functions (q odd)

• Planar functions ⇒ commutative (pre)semiﬁelds? (Conjectured by Dembowksi and Ostrom 1968)

3k +1 • Counter-example: (1997) Coulter-Matthews monomials x 2 over F3n . • Other examples? p > 3?

• By a planar function f over Fq, one can always deﬁne an aﬃne plane Πf using D = {(x, f (x)) : x ∈ Fq}.

Points: (x, y) ∈ Fq × Fq

Lines: `a,b = D + (a, b) = {(x + a, f (x) + b): x ∈ Fq}, `a = {(a, y): y ∈ Fq}.

16/35 • Coulter-Matthews planar functions deﬁne a very special plane (Lenz-Barlotti II.1).

• In general, Πf is a semifield plane ⇒ f is DO+affine? • Dembowski and Ostrom gave a criterion (1968). • Dempwolff and Röder investigated the case in which f is a monomial (2006). • Partially answered by Coulter and Henderson (2008). Theorem (Z., 2018)

The plane Πf is a commutative semiﬁeld plane if and only if f is a DO+aﬃne polynomial.

Planar functions (q odd)

• When f is a DO+aﬃne polynomial, i.e. P pi +pj P pi f (x) = aij x + ci x ,Πf is a semiﬁeld plane.

17/35 • In general, Πf is a semifield plane ⇒ f is DO+affine? • Dembowski and Ostrom gave a criterion (1968). • Dempwolff and Röder investigated the case in which f is a monomial (2006). • Partially answered by Coulter and Henderson (2008). Theorem (Z., 2018)

The plane Πf is a commutative semiﬁeld plane if and only if f is a DO+aﬃne polynomial.

Planar functions (q odd)

• When f is a DO+affine polynomial, i.e. P pi +pj P pi f (x) = aij x + ci x ,Πf is a semifield plane. • Coulter-Matthews planar functions define a very special plane (Lenz-Barlotti II.1).

17/35 • Dembowski and Ostrom gave a criterion (1968). • Dempwolﬀ and R¨oder investigated the case in which f is a monomial (2006). • Partially answered by Coulter and Henderson (2008). Theorem (Z., 2018)

The plane Πf is a commutative semiﬁeld plane if and only if f is a DO+aﬃne polynomial.

Planar functions (q odd)

• When f is a DO+affine polynomial, i.e. P pi +pj P pi f (x) = aij x + ci x ,Πf is a semifield plane. • Coulter-Matthews planar functions define a very special plane (Lenz-Barlotti II.1).

• In general, Πf is a semiﬁeld plane ⇒ f is DO+aﬃne?

17/35 • Dempwolﬀ and R¨oder investigated the case in which f is a monomial (2006). • Partially answered by Coulter and Henderson (2008). Theorem (Z., 2018)

The plane Πf is a commutative semiﬁeld plane if and only if f is a DO+aﬃne polynomial.

Planar functions (q odd)

• When f is a DO+affine polynomial, i.e. P pi +pj P pi f (x) = aij x + ci x ,Πf is a semifield plane. • Coulter-Matthews planar functions define a very special plane (Lenz-Barlotti II.1).

• In general, Πf is a semiﬁeld plane ⇒ f is DO+aﬃne? • Dembowski and Ostrom gave a criterion (1968).

17/35 • Partially answered by Coulter and Henderson (2008). Theorem (Z., 2018)

The plane Πf is a commutative semiﬁeld plane if and only if f is a DO+aﬃne polynomial.

Planar functions (q odd)

• When f is a DO+affine polynomial, i.e. P pi +pj P pi f (x) = aij x + ci x ,Πf is a semifield plane. • Coulter-Matthews planar functions define a very special plane (Lenz-Barlotti II.1).

• In general, Πf is a semifield plane ⇒ f is DO+affine? • Dembowski and Ostrom gave a criterion (1968). • Dempwolff and Röder investigated the case in which f is a monomial (2006).

17/35 Theorem (Z., 2018)

The plane Πf is a commutative semiﬁeld plane if and only if f is a DO+aﬃne polynomial.

Planar functions (q odd)

• When f is a DO+affine polynomial, i.e. P pi +pj P pi f (x) = aij x + ci x ,Πf is a semifield plane. • Coulter-Matthews planar functions define a very special plane (Lenz-Barlotti II.1).

17/35 Planar functions (q odd)

• When f is a DO+affine polynomial, i.e. P pi +pj P pi f (x) = aij x + ci x ,Πf is a semifield plane. • Coulter-Matthews planar functions define a very special plane (Lenz-Barlotti II.1).

The plane Πf is a commutative semiﬁeld plane if and only if f is a DO+aﬃne polynomial. 17/35 • q even, a − b = a + b

• If f (x0 + a) + f (x0) = b, then f ((x0 + a) + a) + f (x0 + a) = b.

• x0 and x0 + a are both mapped to b. Not a permutation. • Hence the previous deﬁnition of planar functions does not work for even q.

• However, for every semiﬁeld S of order q, there exits a group 2 (G, ·) of order q and D ⊆ G of size q deﬁning a plane ΠD : Points: g ∈ G

Lines: `a = aD for a ∈ G and the cosets of H E G which is of size q.

Planar functions (q even)

• Recall that f is planar on Fq if x 7→ f (x + a) − f (x) is a permutation for every a 6= 0.

18/35 , a − b = a + b

• If f (x0 + a) + f (x0) = b, then f ((x0 + a) + a) + f (x0 + a) = b.

• x0 and x0 + a are both mapped to b. Not a permutation. • Hence the previous deﬁnition of planar functions does not work for even q.

• However, for every semiﬁeld S of order q, there exits a group 2 (G, ·) of order q and D ⊆ G of size q deﬁning a plane ΠD : Points: g ∈ G

Lines: `a = aD for a ∈ G and the cosets of H E G which is of size q.

Planar functions (q even)

• Recall that f is planar on Fq if x 7→ f (x + a) − f (x) is a permutation for every a 6= 0. • q even

18/35 • If f (x0 + a) + f (x0) = b, then f ((x0 + a) + a) + f (x0 + a) = b.

• x0 and x0 + a are both mapped to b. Not a permutation. • Hence the previous deﬁnition of planar functions does not work for even q.

• However, for every semiﬁeld S of order q, there exits a group 2 (G, ·) of order q and D ⊆ G of size q deﬁning a plane ΠD : Points: g ∈ G

Lines: `a = aD for a ∈ G and the cosets of H E G which is of size q.

Planar functions (q even)

• Recall that f is planar on Fq if x 7→ f (x + a) − f (x) is a permutation for every a 6= 0. • q even, a − b = a + b

18/35 then f ((x0 + a) + a) + f (x0 + a) = b.

• x0 and x0 + a are both mapped to b. Not a permutation. • Hence the previous deﬁnition of planar functions does not work for even q.

• However, for every semiﬁeld S of order q, there exits a group 2 (G, ·) of order q and D ⊆ G of size q deﬁning a plane ΠD : Points: g ∈ G

Lines: `a = aD for a ∈ G and the cosets of H E G which is of size q.

Planar functions (q even)

• Recall that f is planar on Fq if x 7→ f (x + a) − f (x) is a permutation for every a 6= 0. • q even, a − b = a + b

• If f (x0 + a) + f (x0) = b,

18/35 • x0 and x0 + a are both mapped to b. Not a permutation. • Hence the previous deﬁnition of planar functions does not work for even q.

• However, for every semiﬁeld S of order q, there exits a group 2 (G, ·) of order q and D ⊆ G of size q deﬁning a plane ΠD : Points: g ∈ G

Lines: `a = aD for a ∈ G and the cosets of H E G which is of size q.

Planar functions (q even)

• Recall that f is planar on Fq if x 7→ f (x + a) − f (x) is a permutation for every a 6= 0. • q even, a − b = a + b

• If f (x0 + a) + f (x0) = b, then f ((x0 + a) + a) + f (x0 + a) = b.

18/35 Not a permutation. • Hence the previous deﬁnition of planar functions does not work for even q.

• However, for every semiﬁeld S of order q, there exits a group 2 (G, ·) of order q and D ⊆ G of size q deﬁning a plane ΠD : Points: g ∈ G

Lines: `a = aD for a ∈ G and the cosets of H E G which is of size q.

Planar functions (q even)

• Recall that f is planar on Fq if x 7→ f (x + a) − f (x) is a permutation for every a 6= 0. • q even, a − b = a + b

• If f (x0 + a) + f (x0) = b, then f ((x0 + a) + a) + f (x0 + a) = b.

• x0 and x0 + a are both mapped to b.

18/35 • Hence the previous deﬁnition of planar functions does not work for even q.

• However, for every semiﬁeld S of order q, there exits a group 2 (G, ·) of order q and D ⊆ G of size q deﬁning a plane ΠD : Points: g ∈ G

Lines: `a = aD for a ∈ G and the cosets of H E G which is of size q.

Planar functions (q even)

• Recall that f is planar on Fq if x 7→ f (x + a) − f (x) is a permutation for every a 6= 0. • q even, a − b = a + b

• If f (x0 + a) + f (x0) = b, then f ((x0 + a) + a) + f (x0 + a) = b.

• x0 and x0 + a are both mapped to b. Not a permutation.

18/35 • However, for every semiﬁeld S of order q, there exits a group 2 (G, ·) of order q and D ⊆ G of size q deﬁning a plane ΠD : Points: g ∈ G

Lines: `a = aD for a ∈ G and the cosets of H E G which is of size q.

Planar functions (q even)

• Recall that f is planar on Fq if x 7→ f (x + a) − f (x) is a permutation for every a 6= 0. • q even, a − b = a + b

• If f (x0 + a) + f (x0) = b, then f ((x0 + a) + a) + f (x0 + a) = b.

• x0 and x0 + a are both mapped to b. Not a permutation. • Hence the previous deﬁnition of planar functions does not work for even q.

18/35 Points: g ∈ G

Lines: `a = aD for a ∈ G and the cosets of H E G which is of size q.

Planar functions (q even)

• Recall that f is planar on Fq if x 7→ f (x + a) − f (x) is a permutation for every a 6= 0. • q even, a − b = a + b

• If f (x0 + a) + f (x0) = b, then f ((x0 + a) + a) + f (x0 + a) = b.

• x0 and x0 + a are both mapped to b. Not a permutation. • Hence the previous deﬁnition of planar functions does not work for even q.

• However, for every semiﬁeld S of order q, there exits a group 2 (G, ·) of order q and D ⊆ G of size q deﬁning a plane ΠD :

18/35 Lines: `a = aD for a ∈ G and the cosets of H E G which is of size q.

Planar functions (q even)

• Recall that f is planar on Fq if x 7→ f (x + a) − f (x) is a permutation for every a 6= 0. • q even, a − b = a + b

• If f (x0 + a) + f (x0) = b, then f ((x0 + a) + a) + f (x0 + a) = b.

• x0 and x0 + a are both mapped to b. Not a permutation. • Hence the previous deﬁnition of planar functions does not work for even q.

• However, for every semiﬁeld S of order q, there exits a group 2 (G, ·) of order q and D ⊆ G of size q deﬁning a plane ΠD : Points: g ∈ G

18/35 Planar functions (q even)

• Recall that f is planar on Fq if x 7→ f (x + a) − f (x) is a permutation for every a 6= 0. • q even, a − b = a + b

• If f (x0 + a) + f (x0) = b, then f ((x0 + a) + a) + f (x0 + a) = b.

• x0 and x0 + a are both mapped to b. Not a permutation. • Hence the previous deﬁnition of planar functions does not work for even q.

• However, for every semiﬁeld S of order q, there exits a group 2 (G, ·) of order q and D ⊆ G of size q deﬁning a plane ΠD : Points: g ∈ G

Lines: `a = aD for a ∈ G and the cosets of H E G which is of size q. 18/35 2 G = (Fq, +), D = {(x, x ∗ x): x ∈ Fq} and H = {(0, y): y ∈ Fq}; m m • when S is commutative and q = 2 , G = C4 , m ∼ m H = 2C4 = F2 and D = ? Deﬁnition

A function f : F2m → F2m is (modiﬁed) planar if x 7→ f (x + a) + f (x) + xa is a permutation of F2m for each ∗ a ∈ F2m .

m • Represent C4 as F2m × F2m with the group operation (x, y) ? (x0, y 0) = (x + x0, y + y 0 + x · x0).

Planar functions (q even)

For instance,

• when S is commutative and q is odd,

19/35 D = {(x, x ∗ x): x ∈ Fq} and H = {(0, y): y ∈ Fq}; m m • when S is commutative and q = 2 , G = C4 , m ∼ m H = 2C4 = F2 and D = ? Deﬁnition

A function f : F2m → F2m is (modiﬁed) planar if x 7→ f (x + a) + f (x) + xa is a permutation of F2m for each ∗ a ∈ F2m .

m • Represent C4 as F2m × F2m with the group operation (x, y) ? (x0, y 0) = (x + x0, y + y 0 + x · x0).

Planar functions (q even)

For instance,

2 • when S is commutative and q is odd, G = (Fq, +),

19/35 m m • when S is commutative and q = 2 , G = C4 , m ∼ m H = 2C4 = F2 and D = ? Deﬁnition

A function f : F2m → F2m is (modiﬁed) planar if x 7→ f (x + a) + f (x) + xa is a permutation of F2m for each ∗ a ∈ F2m .

m • Represent C4 as F2m × F2m with the group operation (x, y) ? (x0, y 0) = (x + x0, y + y 0 + x · x0).

Planar functions (q even)

For instance,

2 • when S is commutative and q is odd, G = (Fq, +), D = {(x, x ∗ x): x ∈ Fq} and H = {(0, y): y ∈ Fq};

19/35 m G = C4 , m ∼ m H = 2C4 = F2 and D = ? Deﬁnition

A function f : F2m → F2m is (modiﬁed) planar if x 7→ f (x + a) + f (x) + xa is a permutation of F2m for each ∗ a ∈ F2m .

m • Represent C4 as F2m × F2m with the group operation (x, y) ? (x0, y 0) = (x + x0, y + y 0 + x · x0).

Planar functions (q even)

For instance,

2 • when S is commutative and q is odd, G = (Fq, +), D = {(x, x ∗ x): x ∈ Fq} and H = {(0, y): y ∈ Fq}; m • when S is commutative and q = 2 ,

19/35 m ∼ m H = 2C4 = F2 and D = ? Deﬁnition

A function f : F2m → F2m is (modiﬁed) planar if x 7→ f (x + a) + f (x) + xa is a permutation of F2m for each ∗ a ∈ F2m .

m • Represent C4 as F2m × F2m with the group operation (x, y) ? (x0, y 0) = (x + x0, y + y 0 + x · x0).

Planar functions (q even)

For instance,

2 • when S is commutative and q is odd, G = (Fq, +), D = {(x, x ∗ x): x ∈ Fq} and H = {(0, y): y ∈ Fq}; m m • when S is commutative and q = 2 , G = C4 ,

19/35 and D = ? Deﬁnition

A function f : F2m → F2m is (modiﬁed) planar if x 7→ f (x + a) + f (x) + xa is a permutation of F2m for each ∗ a ∈ F2m .

m • Represent C4 as F2m × F2m with the group operation (x, y) ? (x0, y 0) = (x + x0, y + y 0 + x · x0).

Planar functions (q even)

For instance,

2 • when S is commutative and q is odd, G = (Fq, +), D = {(x, x ∗ x): x ∈ Fq} and H = {(0, y): y ∈ Fq}; m m • when S is commutative and q = 2 , G = C4 , m ∼ m H = 2C4 = F2

19/35 Deﬁnition

A function f : F2m → F2m is (modiﬁed) planar if x 7→ f (x + a) + f (x) + xa is a permutation of F2m for each ∗ a ∈ F2m .

m • Represent C4 as F2m × F2m with the group operation (x, y) ? (x0, y 0) = (x + x0, y + y 0 + x · x0).

Planar functions (q even)

For instance,

2 • when S is commutative and q is odd, G = (Fq, +), D = {(x, x ∗ x): x ∈ Fq} and H = {(0, y): y ∈ Fq}; m m • when S is commutative and q = 2 , G = C4 , m ∼ m H = 2C4 = F2 and D = ?

19/35 m • Represent C4 as F2m × F2m with the group operation (x, y) ? (x0, y 0) = (x + x0, y + y 0 + x · x0).

Planar functions (q even)

For instance,

2 • when S is commutative and q is odd, G = (Fq, +), D = {(x, x ∗ x): x ∈ Fq} and H = {(0, y): y ∈ Fq}; m m • when S is commutative and q = 2 , G = C4 , m ∼ m H = 2C4 = F2 and D = ? Deﬁnition

A function f : F2m → F2m is (modiﬁed) planar if x 7→ f (x + a) + f (x) + xa is a permutation of F2m for each ∗ a ∈ F2m .

19/35 Planar functions (q even)

For instance,

A function f : F2m → F2m is (modiﬁed) planar if x 7→ f (x + a) + f (x) + xa is a permutation of F2m for each ∗ a ∈ F2m .

m • Represent C4 as F2m × F2m with the group operation (x, y) ? (x0, y 0) = (x + x0, y + y 0 + x · x0).

19/35 Planar functions (q even)

For instance,

2 • when S is commutative and q is odd, G = (Fq, +), D = {(x, x ∗ x): x ∈ Fq} and H = {(0, y): y ∈ Fq}; m m • when S is commutative and q = 2 , G = C4 , m ∼ m H = 2C4 = F2 and D = {(x, f (x)) : x ∈ F2n } Deﬁnition

A function f : F2m → F2m is (modiﬁed) planar if x 7→ f (x + a) + f (x) + xa is a permutation of F2m for each ∗ a ∈ F2m .

m • Represent C4 as F2m × F2m with the group operation (x, y) ? (x0, y 0) = (x + x0, y + y 0 + x · x0).

19/35 there are many planar functions over Fq. • Only one family is known. • No example of order 22k . For instance q = 28. • Question: Are there any non-DO planar polynomials, like the Coulter-Matthews planar functions for q odd? • Related to a long standing open problem in coding theory.

Planar functions (q even)

• As there are exponentially many commutative semiﬁelds of even order q (Kantor 2003),

20/35 • Only one family is known. • No example of order 22k . For instance q = 28. • Question: Are there any non-DO planar polynomials, like the Coulter-Matthews planar functions for q odd? • Related to a long standing open problem in coding theory.

Planar functions (q even)

• As there are exponentially many commutative semiﬁelds of even order q (Kantor 2003), there are many planar functions over Fq.

20/35 • No example of order 22k . For instance q = 28. • Question: Are there any non-DO planar polynomials, like the Coulter-Matthews planar functions for q odd? • Related to a long standing open problem in coding theory.

Planar functions (q even)

• As there are exponentially many commutative semiﬁelds of even order q (Kantor 2003), there are many planar functions over Fq. • Only one family is known.

20/35 For instance q = 28. • Question: Are there any non-DO planar polynomials, like the Coulter-Matthews planar functions for q odd? • Related to a long standing open problem in coding theory.

Planar functions (q even)

• As there are exponentially many commutative semiﬁelds of even order q (Kantor 2003), there are many planar functions over Fq. • Only one family is known. • No example of order 22k .

20/35 • Question: Are there any non-DO planar polynomials, like the Coulter-Matthews planar functions for q odd? • Related to a long standing open problem in coding theory.

Planar functions (q even)

20/35 • Related to a long standing open problem in coding theory.

Planar functions (q even)

• As there are exponentially many commutative semiﬁelds of even order q (Kantor 2003), there are many planar functions over Fq. • Only one family is known. • No example of order 22k . For instance q = 28. • Question: Are there any non-DO planar polynomials, like the Coulter-Matthews planar functions for q odd?

20/35 • The plane Πf is a commutative semifield plane if and only if f is a DO+affine polynomial (Z., 2013). • Using algebraic geometry, one can get classification results of them (Schmidt, Z. 2013, Müller,Zieve 2015, Bartoli, Schmidt 2019).

Planar functions (q even)

20/35 • Using algebraic geometry, one can get classiﬁcation results of them (Schmidt, Z. 2013, M¨uller,Zieve 2015, Bartoli, Schmidt 2019).

Planar functions (q even)

• The plane Πf is a commutative semiﬁeld plane if and only if f is a DO+aﬃne polynomial (Z., 2013).

20/35 Planar functions (q even)

• The plane Πf is a commutative semifield plane if and only if f is a DO+affine polynomial (Z., 2013). • Using algebraic geometry, one can get classification results of them (Schmidt, Z. 2013, Müller,Zieve 2015, Bartoli, Schmidt 2019). 20/35 Maximum Rank-Distance Codes C is an MRD code of n×n minimum distance n in Fp , and La + Lb = La+b.

n×n • La is Fp-linear, can be viewed as a matrix in Fp

• La − Lb is invertible for a 6= b; n •C = {La : a ∈ Fpn } has p elements.

Deﬁnition n×n Given C ⊆ Fq , if • #C = qnk , • for each distinct M, N ∈ C, rank(M − N) ≥ n − k + 1,

then C is a maximum rank-distance (MRD, for short) codes in n×n Fq . Its minimum distance d = n − k + 1.

MRD Codes

Let (Fpn , +, ∗) be a semiﬁeld. Deﬁne La : x 7→ x ∗ a.

21/35 C is an MRD code of n×n minimum distance n in Fp , and La + Lb = La+b.

• La − Lb is invertible for a 6= b; n •C = {La : a ∈ Fpn } has p elements.

Deﬁnition n×n Given C ⊆ Fq , if • #C = qnk , • for each distinct M, N ∈ C, rank(M − N) ≥ n − k + 1,

then C is a maximum rank-distance (MRD, for short) codes in n×n Fq . Its minimum distance d = n − k + 1.

MRD Codes

Let (Fpn , +, ∗) be a semiﬁeld. Deﬁne La : x 7→ x ∗ a. n×n • La is Fp-linear, can be viewed as a matrix in Fp

21/35 C is an MRD code of n×n minimum distance n in Fp , and La + Lb = La+b.

n •C = {La : a ∈ Fpn } has p elements.

Deﬁnition n×n Given C ⊆ Fq , if • #C = qnk , • for each distinct M, N ∈ C, rank(M − N) ≥ n − k + 1,

then C is a maximum rank-distance (MRD, for short) codes in n×n Fq . Its minimum distance d = n − k + 1.

MRD Codes

Let (Fpn , +, ∗) be a semiﬁeld. Deﬁne La : x 7→ x ∗ a. n×n • La is Fp-linear, can be viewed as a matrix in Fp

• La − Lb is invertible for a 6= b;

21/35 C is an MRD code of n×n minimum distance n in Fp , and La + Lb = La+b. Deﬁnition n×n Given C ⊆ Fq , if • #C = qnk , • for each distinct M, N ∈ C, rank(M − N) ≥ n − k + 1,

then C is a maximum rank-distance (MRD, for short) codes in n×n Fq . Its minimum distance d = n − k + 1.

MRD Codes

Let (Fpn , +, ∗) be a semiﬁeld. Deﬁne La : x 7→ x ∗ a. n×n • La is Fp-linear, can be viewed as a matrix in Fp

• La − Lb is invertible for a 6= b; n •C = {La : a ∈ Fpn } has p elements.

21/35 C is an MRD code of n×n minimum distance n in Fp , and La + Lb = La+b.

• #C = qnk , • for each distinct M, N ∈ C, rank(M − N) ≥ n − k + 1,

then C is a maximum rank-distance (MRD, for short) codes in n×n Fq . Its minimum distance d = n − k + 1.

MRD Codes

Let (Fpn , +, ∗) be a semiﬁeld. Deﬁne La : x 7→ x ∗ a. n×n • La is Fp-linear, can be viewed as a matrix in Fp

• La − Lb is invertible for a 6= b; n •C = {La : a ∈ Fpn } has p elements.

Deﬁnition n×n Given C ⊆ Fq , if

21/35 C is an MRD code of n×n minimum distance n in Fp , and La + Lb = La+b.

• for each distinct M, N ∈ C, rank(M − N) ≥ n − k + 1,

then C is a maximum rank-distance (MRD, for short) codes in n×n Fq . Its minimum distance d = n − k + 1.

MRD Codes

Let (Fpn , +, ∗) be a semiﬁeld. Deﬁne La : x 7→ x ∗ a. n×n • La is Fp-linear, can be viewed as a matrix in Fp

• La − Lb is invertible for a 6= b; n •C = {La : a ∈ Fpn } has p elements.

Deﬁnition n×n Given C ⊆ Fq , if • #C = qnk ,

21/35 C is an MRD code of n×n minimum distance n in Fp , and La + Lb = La+b.

then C is a maximum rank-distance (MRD, for short) codes in n×n Fq . Its minimum distance d = n − k + 1.

MRD Codes

Let (Fpn , +, ∗) be a semiﬁeld. Deﬁne La : x 7→ x ∗ a. n×n • La is Fp-linear, can be viewed as a matrix in Fp

• La − Lb is invertible for a 6= b; n •C = {La : a ∈ Fpn } has p elements.

Deﬁnition n×n Given C ⊆ Fq , if • #C = qnk , • for each distinct M, N ∈ C, rank(M − N) ≥ n − k + 1,

21/35 C is an MRD code of n×n minimum distance n in Fp , and La + Lb = La+b.

Its minimum distance d = n − k + 1.

MRD Codes

Let (Fpn , +, ∗) be a semiﬁeld. Deﬁne La : x 7→ x ∗ a. n×n • La is Fp-linear, can be viewed as a matrix in Fp

• La − Lb is invertible for a 6= b; n •C = {La : a ∈ Fpn } has p elements.

Deﬁnition n×n Given C ⊆ Fq , if • #C = qnk , • for each distinct M, N ∈ C, rank(M − N) ≥ n − k + 1,

then C is a maximum rank-distance (MRD, for short) codes in n×n Fq . 21/35 C is an MRD code of n×n minimum distance n in Fp , and La + Lb = La+b.

MRD Codes

Let (Fpn , +, ∗) be a semiﬁeld. Deﬁne La : x 7→ x ∗ a. n×n • La is Fp-linear, can be viewed as a matrix in Fp

• La − Lb is invertible for a 6= b; n •C = {La : a ∈ Fpn } has p elements.

Deﬁnition n×n Given C ⊆ Fq , if • #C = qnk , • for each distinct M, N ∈ C, rank(M − N) ≥ n − k + 1,

then C is a maximum rank-distance (MRD, for short) codes in n×n Fq . Its minimum distance d = n − k + 1. 21/35 and La + Lb = La+b.

MRD Codes

Let (Fpn , +, ∗) be a semiﬁeld. Deﬁne La : x 7→ x ∗ a. n×n • La is Fp-linear, can be viewed as a matrix in Fp

• La − Lb is invertible for a 6= b; n •C = {La : a ∈ Fpn } has p elements. C is an MRD code of n×n minimum distance n in Fp , Deﬁnition n×n Given C ⊆ Fq , if • #C = qnk , • for each distinct M, N ∈ C, rank(M − N) ≥ n − k + 1,

then C is a maximum rank-distance (MRD, for short) codes in n×n Fq . Its minimum distance d = n − k + 1. 21/35 MRD Codes

Let (Fpn , +, ∗) be a semiﬁeld. Deﬁne La : x 7→ x ∗ a. n×n • La is Fp-linear, can be viewed as a matrix in Fp

• La − Lb is invertible for a 6= b; n •C = {La : a ∈ Fpn } has p elements. C is an MRD code of n×n minimum distance n in Fp , and La + Lb = La+b. Deﬁnition n×n Given C ⊆ Fq , if • #C = qnk , • for each distinct M, N ∈ C, rank(M − N) ≥ n − k + 1,

then C is a maximum rank-distance (MRD, for short) codes in n×n Fq . Its minimum distance d = n − k + 1. 21/35 if there are m×n A ∈ GL(m, K), B ∈ GL(n, K), C ∈ K and γ ∈ Aut(K) such that γ C2 = {AX B + C : X ∈ C1},

γ γ where X := (xij ).

m×n • (A, B, C, γ) is an isometry over K . • When m = n, every isometry is AX γB + C or A(X γ)T B + C.

• If C1 and C2 are linear over K, then we can assume that C = O.

Equivalence

Deﬁnition m×n Given C1 and C2 ⊆ K , they are equivalent

22/35 γ C2 = {AX B + C : X ∈ C1},

γ γ where X := (xij ).

m×n • (A, B, C, γ) is an isometry over K . • When m = n, every isometry is AX γB + C or A(X γ)T B + C.

• If C1 and C2 are linear over K, then we can assume that C = O.

Equivalence

Deﬁnition m×n Given C1 and C2 ⊆ K , they are equivalent if there are m×n A ∈ GL(m, K), B ∈ GL(n, K), C ∈ K and γ ∈ Aut(K) such that

22/35 m×n • (A, B, C, γ) is an isometry over K . • When m = n, every isometry is AX γB + C or A(X γ)T B + C.

• If C1 and C2 are linear over K, then we can assume that C = O.

Equivalence

Deﬁnition m×n Given C1 and C2 ⊆ K , they are equivalent if there are m×n A ∈ GL(m, K), B ∈ GL(n, K), C ∈ K and γ ∈ Aut(K) such that γ C2 = {AX B + C : X ∈ C1},

γ γ where X := (xij ).

22/35 • When m = n, every isometry is AX γB + C or A(X γ)T B + C.

• If C1 and C2 are linear over K, then we can assume that C = O.

Equivalence

Deﬁnition m×n Given C1 and C2 ⊆ K , they are equivalent if there are m×n A ∈ GL(m, K), B ∈ GL(n, K), C ∈ K and γ ∈ Aut(K) such that γ C2 = {AX B + C : X ∈ C1},

γ γ where X := (xij ).

m×n • (A, B, C, γ) is an isometry over K .

22/35 • If C1 and C2 are linear over K, then we can assume that C = O.

Equivalence

Deﬁnition m×n Given C1 and C2 ⊆ K , they are equivalent if there are m×n A ∈ GL(m, K), B ∈ GL(n, K), C ∈ K and γ ∈ Aut(K) such that γ C2 = {AX B + C : X ∈ C1},

γ γ where X := (xij ).

m×n • (A, B, C, γ) is an isometry over K . • When m = n, every isometry is AX γB + C or A(X γ)T B + C.

22/35 Equivalence

Deﬁnition m×n Given C1 and C2 ⊆ K , they are equivalent if there are m×n A ∈ GL(m, K), B ∈ GL(n, K), C ∈ K and γ ∈ Aut(K) such that γ C2 = {AX B + C : X ∈ C1},

γ γ where X := (xij ).

m×n • (A, B, C, γ) is an isometry over K . • When m = n, every isometry is AX γB + C or A(X γ)T B + C.

• If C1 and C2 are linear over K, then we can assume that C = O.

22/35 How to construct MRD codes of minimum distance d < n? • Gabidulin codes (k = n − d + 1) (Delsarte 1978), (Gabidulin 1985) q qk−1 Gk = {a0X + a1X + ... ak−1X : a0, a1,..., ak−1 ∈ Fqn }. . For each f ∈ G, f has at most qk−1 roots. . #G = qnk = qn(n−d+1) with d = n − k + 1.

•G k is MRD. • One may take σ = qt with gcd(t, n) = 1 and replace q by σ. • More constructions?

MRD Codes

A semiﬁeld is just a linear MRD code of minimum distance n in n×n Fq , where Fq is one of its nucleus.

23/35 • Gabidulin codes (k = n − d + 1) (Delsarte 1978), (Gabidulin 1985) q qk−1 Gk = {a0X + a1X + ... ak−1X : a0, a1,..., ak−1 ∈ Fqn }. . For each f ∈ G, f has at most qk−1 roots. . #G = qnk = qn(n−d+1) with d = n − k + 1.

•G k is MRD. • One may take σ = qt with gcd(t, n) = 1 and replace q by σ. • More constructions?

MRD Codes

A semiﬁeld is just a linear MRD code of minimum distance n in n×n Fq , where Fq is one of its nucleus.

How to construct MRD codes of minimum distance d < n?

23/35 . For each f ∈ G, f has at most qk−1 roots. . #G = qnk = qn(n−d+1) with d = n − k + 1.

•G k is MRD. • One may take σ = qt with gcd(t, n) = 1 and replace q by σ. • More constructions?

MRD Codes

A semiﬁeld is just a linear MRD code of minimum distance n in n×n Fq , where Fq is one of its nucleus.

How to construct MRD codes of minimum distance d < n? • Gabidulin codes (k = n − d + 1) (Delsarte 1978), (Gabidulin 1985) q qk−1 Gk = {a0X + a1X + ... ak−1X : a0, a1,..., ak−1 ∈ Fqn }.

23/35 . #G = qnk = qn(n−d+1) with d = n − k + 1.

•G k is MRD. • One may take σ = qt with gcd(t, n) = 1 and replace q by σ. • More constructions?

MRD Codes

A semiﬁeld is just a linear MRD code of minimum distance n in n×n Fq , where Fq is one of its nucleus.

How to construct MRD codes of minimum distance d < n? • Gabidulin codes (k = n − d + 1) (Delsarte 1978), (Gabidulin 1985) q qk−1 Gk = {a0X + a1X + ... ak−1X : a0, a1,..., ak−1 ∈ Fqn }. . For each f ∈ G, f has at most qk−1 roots.

23/35 •G k is MRD. • One may take σ = qt with gcd(t, n) = 1 and replace q by σ. • More constructions?

MRD Codes

A semiﬁeld is just a linear MRD code of minimum distance n in n×n Fq , where Fq is one of its nucleus.

23/35 • One may take σ = qt with gcd(t, n) = 1 and replace q by σ. • More constructions?

MRD Codes

A semiﬁeld is just a linear MRD code of minimum distance n in n×n Fq , where Fq is one of its nucleus.

•G k is MRD.

23/35 • More constructions?

MRD Codes

A semiﬁeld is just a linear MRD code of minimum distance n in n×n Fq , where Fq is one of its nucleus.

•G k is MRD. • One may take σ = qt with gcd(t, n) = 1 and replace q by σ.

23/35 • More constructions?

MRD Codes

A semiﬁeld is just a linear MRD code of minimum distance n in n×n Fq , where Fq is one of its nucleus.

How to construct MRD codes of minimum distance d < n? • Gabidulin codes (k = n − d + 1) (Delsarte 1978), (Gabidulin 1985) σ σk−1 Gk = {a0X + a1X + ... ak−1X : a0, a1,..., ak−1 ∈ Fqn }. . For each f ∈ G, f has at most qk−1 roots. . #G = qnk = qn(n−d+1) with d = n − k + 1.

•G k is MRD. • One may take σ = qt with gcd(t, n) = 1 and replace q by σ.

23/35 MRD Codes

A semiﬁeld is just a linear MRD code of minimum distance n in n×n Fq , where Fq is one of its nucleus.

•G k is MRD. • One may take σ = qt with gcd(t, n) = 1 and replace q by σ. • More constructions?

23/35 • Scattered linear sets on a line (k = 2)

• Moore matrices

Besides, there are also two special nonlinear constructions of MRD codes.

• Durante and Siciliano 2017 generalizing Cossidente, Marino and Pavese 2016.

• Otal and Ozbudak¨ 2018 (analogue of nearﬁelds).

More MRD Codes

There are mainly three approaches:

• Twisting

24/35 • Moore matrices

Besides, there are also two special nonlinear constructions of MRD codes.

• Durante and Siciliano 2017 generalizing Cossidente, Marino and Pavese 2016.

• Otal and Ozbudak¨ 2018 (analogue of nearﬁelds).

More MRD Codes

There are mainly three approaches:

• Twisting

• Scattered linear sets on a line (k = 2)

24/35 Besides, there are also two special nonlinear constructions of MRD codes.

• Durante and Siciliano 2017 generalizing Cossidente, Marino and Pavese 2016.

• Otal and Ozbudak¨ 2018 (analogue of nearﬁelds).

More MRD Codes

There are mainly three approaches:

• Twisting

• Scattered linear sets on a line (k = 2)

• Moore matrices

24/35 • Durante and Siciliano 2017 generalizing Cossidente, Marino and Pavese 2016.

• Otal and Ozbudak¨ 2018 (analogue of nearﬁelds).

More MRD Codes

There are mainly three approaches:

• Twisting

• Scattered linear sets on a line (k = 2)

• Moore matrices

Besides, there are also two special nonlinear constructions of MRD codes.

24/35 • Otal and Ozbudak¨ 2018 (analogue of nearﬁelds).

More MRD Codes

There are mainly three approaches:

• Twisting

• Scattered linear sets on a line (k = 2)

• Moore matrices

Besides, there are also two special nonlinear constructions of MRD codes.

• Durante and Siciliano 2017 generalizing Cossidente, Marino and Pavese 2016.

24/35 More MRD Codes

There are mainly three approaches:

• Twisting

• Scattered linear sets on a line (k = 2)

• Moore matrices

Besides, there are also two special nonlinear constructions of MRD codes.

• Durante and Siciliano 2017 generalizing Cossidente, Marino and Pavese 2016.

• Otal and Ozbudak¨ 2018 (analogue of nearﬁelds).

24/35 Twisted Delsarte-Gabidulin codes (Sheekey, 2016)

σ σk−1 qh σk qn {a0X + a1X + ... ak−1X + ηa0 X : ai ∈ Fqn } (mod X − X ).

Note that if η = 0, it is a Delsarte-Gabidulin code.

For even n (Trombetti and Z., 2019)

σ σk−1 σk ∗ qn {aX +a1X +... ak−1X +ηbX : ai ∈ Fqn , a, b ∈ Fqn/2 } (mod X −X ). (Twisted) cyclic construction (Sheekey, arixv)

n {· · · } mod r f (X ) (in Fqn [X ; σ]),

where f is irreducible in Fqn [X ; σ].

Twisting

Let σ = qt where gcd(t, n) = 1.

25/35 σ σk−1 qh σk qn {a0X + a1X + ... ak−1X + ηa0 X : ai ∈ Fqn } (mod X − X ).

Note that if η = 0, it is a Delsarte-Gabidulin code.

For even n (Trombetti and Z., 2019)

σ σk−1 σk ∗ qn {aX +a1X +... ak−1X +ηbX : ai ∈ Fqn , a, b ∈ Fqn/2 } (mod X −X ). (Twisted) cyclic construction (Sheekey, arixv)

n {· · · } mod r f (X ) (in Fqn [X ; σ]),

where f is irreducible in Fqn [X ; σ].

Twisting

Let σ = qt where gcd(t, n) = 1. Twisted Delsarte-Gabidulin codes (Sheekey, 2016)

25/35 Note that if η = 0, it is a Delsarte-Gabidulin code.

For even n (Trombetti and Z., 2019)

σ σk−1 σk ∗ qn {aX +a1X +... ak−1X +ηbX : ai ∈ Fqn , a, b ∈ Fqn/2 } (mod X −X ). (Twisted) cyclic construction (Sheekey, arixv)

n {· · · } mod r f (X ) (in Fqn [X ; σ]),

where f is irreducible in Fqn [X ; σ].

Twisting

Let σ = qt where gcd(t, n) = 1. Twisted Delsarte-Gabidulin codes (Sheekey, 2016)

σ σk−1 qh σk qn {a0X + a1X + ... ak−1X + ηa0 X : ai ∈ Fqn } (mod X − X ).

25/35 For even n (Trombetti and Z., 2019)

σ σk−1 σk ∗ qn {aX +a1X +... ak−1X +ηbX : ai ∈ Fqn , a, b ∈ Fqn/2 } (mod X −X ). (Twisted) cyclic construction (Sheekey, arixv)

n {· · · } mod r f (X ) (in Fqn [X ; σ]),

where f is irreducible in Fqn [X ; σ].

Twisting

Let σ = qt where gcd(t, n) = 1. Twisted Delsarte-Gabidulin codes (Sheekey, 2016)

σ σk−1 qh σk qn {a0X + a1X + ... ak−1X + ηa0 X : ai ∈ Fqn } (mod X − X ).

Note that if η = 0, it is a Delsarte-Gabidulin code.

25/35 σ σk−1 σk ∗ qn {aX +a1X +... ak−1X +ηbX : ai ∈ Fqn , a, b ∈ Fqn/2 } (mod X −X ). (Twisted) cyclic construction (Sheekey, arixv)

n {· · · } mod r f (X ) (in Fqn [X ; σ]),

where f is irreducible in Fqn [X ; σ].

Twisting

Let σ = qt where gcd(t, n) = 1. Twisted Delsarte-Gabidulin codes (Sheekey, 2016)

σ σk−1 qh σk qn {a0X + a1X + ... ak−1X + ηa0 X : ai ∈ Fqn } (mod X − X ).

Note that if η = 0, it is a Delsarte-Gabidulin code.

For even n (Trombetti and Z., 2019)

25/35 (Twisted) cyclic construction (Sheekey, arixv)

n {· · · } mod r f (X ) (in Fqn [X ; σ]),

where f is irreducible in Fqn [X ; σ].

Twisting

Let σ = qt where gcd(t, n) = 1. Twisted Delsarte-Gabidulin codes (Sheekey, 2016)

σ σk−1 qh σk qn {a0X + a1X + ... ak−1X + ηa0 X : ai ∈ Fqn } (mod X − X ).

Note that if η = 0, it is a Delsarte-Gabidulin code.

For even n (Trombetti and Z., 2019)

σ σk−1 σk ∗ qn {aX +a1X +... ak−1X +ηbX : ai ∈ Fqn , a, b ∈ Fqn/2 } (mod X −X ).

25/35 n {· · · } mod r f (X ) (in Fqn [X ; σ]),

where f is irreducible in Fqn [X ; σ].

Twisting

Let σ = qt where gcd(t, n) = 1. Twisted Delsarte-Gabidulin codes (Sheekey, 2016)

σ σk−1 qh σk qn {a0X + a1X + ... ak−1X + ηa0 X : ai ∈ Fqn } (mod X − X ).

Note that if η = 0, it is a Delsarte-Gabidulin code.

For even n (Trombetti and Z., 2019)

σ σk−1 σk ∗ qn {aX +a1X +... ak−1X +ηbX : ai ∈ Fqn , a, b ∈ Fqn/2 } (mod X −X ). (Twisted) cyclic construction (Sheekey, arixv)

25/35 Twisting

Let σ = qt where gcd(t, n) = 1. Twisted Delsarte-Gabidulin codes (Sheekey, 2016)

σ σk−1 qh σk qn {a0X + a1X + ... ak−1X + ηa0 X : ai ∈ Fqn } (mod X − X ).

Note that if η = 0, it is a Delsarte-Gabidulin code.

For even n (Trombetti and Z., 2019)

σ σk−1 σk ∗ qn {aX +a1X +... ak−1X +ηbX : ai ∈ Fqn , a, b ∈ Fqn/2 } (mod X −X ). (Twisted) cyclic construction (Sheekey, arixv)

n {· · · } mod r f (X ) (in Fqn [X ; σ]),

where f is irreducible in Fqn [X ; σ].

25/35 qs q2s 0 qs q(n−1)s {a0X +a1X +ηa0X : a0, a1 ∈ Fqn } = {aX +η bX +bX : a, b ∈ Fqn } (apply (·)q(n−1)s and mod X qn − X )

•F is MRD if and only if

f (x) f (y) y = ⇔ ∈ . x y x Fq

• A polynomial f satisfying the condition is called a scattered polynomial.

Scattered linear sets

n×n Fq-linear MRD codes of d = n − 1 (i.e. k = 2) in Fq has to be in the form F = {aX + bf (X ): a, b ∈ Fqn }.

26/35 •F is MRD if and only if

f (x) f (y) y = ⇔ ∈ . x y x Fq

• A polynomial f satisfying the condition is called a scattered polynomial.

Scattered linear sets

n×n Fq-linear MRD codes of d = n − 1 (i.e. k = 2) in Fq has to be in the form F = {aX + bf (X ): a, b ∈ Fqn }.

qs q2s 0 qs q(n−1)s {a0X +a1X +ηa0X : a0, a1 ∈ Fqn } = {aX +η bX +bX : a, b ∈ Fqn } (apply (·)q(n−1)s and mod X qn − X )

26/35 • A polynomial f satisfying the condition is called a scattered polynomial.

Scattered linear sets

n×n Fq-linear MRD codes of d = n − 1 (i.e. k = 2) in Fq has to be in the form F = {aX + bf (X ): a, b ∈ Fqn }.

qs q2s 0 qs q(n−1)s {a0X +a1X +ηa0X : a0, a1 ∈ Fqn } = {aX +η bX +bX : a, b ∈ Fqn } (apply (·)q(n−1)s and mod X qn − X )

•F is MRD if and only if

f (x) f (y) y = ⇔ ∈ . x y x Fq

26/35 Scattered linear sets

n×n Fq-linear MRD codes of d = n − 1 (i.e. k = 2) in Fq has to be in the form F = {aX + bf (X ): a, b ∈ Fqn }.

qs q2s 0 qs q(n−1)s {a0X +a1X +ηa0X : a0, a1 ∈ Fqn } = {aX +η bX +bX : a, b ∈ Fqn } (apply (·)q(n−1)s and mod X qn − X )

•F is MRD if and only if

f (x) f (y) y = ⇔ ∈ . x y x Fq

• A polynomial f satisfying the condition is called a scattered polynomial.

26/35 • Hence it is equivalent to f (x) f (y) y x = y ⇔ x ∈ Fq. • Constructions for n = 6, 8 (Csajb´ok,Marino, Polverino, Zanella, Zullo). • Classiﬁcation results (Bartoli, Z. 2018) (Bartoli, Montanucci arxiv).

Scattered polynomials

• In ﬁnite geometries, a maximum scattered linear set over PG(1, qn): 2 U = {(x, f (x)) : x ∈ Fqn } ⊆ Fqn , f (x) ∗ L(U)= {hui : u ∈ U \{0}} = 1, : x ∈ n , Fqn x Fq qn − 1 #L(U) = . q − 1

27/35 • Constructions for n = 6, 8 (Csajb´ok,Marino, Polverino, Zanella, Zullo). • Classiﬁcation results (Bartoli, Z. 2018) (Bartoli, Montanucci arxiv).

Scattered polynomials

27/35 • Classiﬁcation results (Bartoli, Z. 2018) (Bartoli, Montanucci arxiv).

Scattered polynomials

• In ﬁnite geometries, a maximum scattered linear set over PG(1, qn): 2 U = {(x, f (x)) : x ∈ Fqn } ⊆ Fqn , f (x) ∗ L(U)= {hui : u ∈ U \{0}} = 1, : x ∈ n , Fqn x Fq qn − 1 #L(U) = . q − 1 • Hence it is equivalent to f (x) f (y) y x = y ⇔ x ∈ Fq. • Constructions for n = 6, 8 (Csajb´ok,Marino, Polverino, Zanella, Zullo).

27/35 Scattered polynomials

arxiv). 27/35 However, for a very special type, this work is quite easy.

qt0 qt1 qtk−1 CΛ := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ],

where Λ = {t0, ··· , tk−1}. Theorem (Csajb´ok,Marino, Polverino, Z, submitted)

Let Λ1 and Λ2 be two k-subsets of {0,..., n − 1}. Deﬁne n qi o C = P a X : a ∈ n for j = 1, 2. Then C and C are j i∈Λj i i Fq 1 2 equivalent if and only if

Λ2 = Λ1 + s := {i + s (mod n): i ∈ Λ1}

for some s ∈ {0, ··· , n − 1}.

Moore matrices

In general, it is quite diﬃcult to tell whether C1 and C2 are equivalent.

28/35 qt0 qt1 qtk−1 CΛ := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ],

where Λ = {t0, ··· , tk−1}. Theorem (Csajb´ok,Marino, Polverino, Z, submitted)

Let Λ1 and Λ2 be two k-subsets of {0,..., n − 1}. Deﬁne n qi o C = P a X : a ∈ n for j = 1, 2. Then C and C are j i∈Λj i i Fq 1 2 equivalent if and only if

Λ2 = Λ1 + s := {i + s (mod n): i ∈ Λ1}

for some s ∈ {0, ··· , n − 1}.

Moore matrices

In general, it is quite diﬃcult to tell whether C1 and C2 are equivalent. However, for a very special type, this work is quite easy.

28/35 Theorem (Csajb´ok,Marino, Polverino, Z, submitted)

Let Λ1 and Λ2 be two k-subsets of {0,..., n − 1}. Deﬁne n qi o C = P a X : a ∈ n for j = 1, 2. Then C and C are j i∈Λj i i Fq 1 2 equivalent if and only if

Λ2 = Λ1 + s := {i + s (mod n): i ∈ Λ1}

for some s ∈ {0, ··· , n − 1}.

Moore matrices

In general, it is quite diﬃcult to tell whether C1 and C2 are equivalent. However, for a very special type, this work is quite easy.

qt0 qt1 qtk−1 CΛ := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ],

where Λ = {t0, ··· , tk−1}.

28/35 Deﬁne n qi o C = P a X : a ∈ n for j = 1, 2. Then C and C are j i∈Λj i i Fq 1 2 equivalent if and only if

Λ2 = Λ1 + s := {i + s (mod n): i ∈ Λ1}

for some s ∈ {0, ··· , n − 1}.

Moore matrices

In general, it is quite diﬃcult to tell whether C1 and C2 are equivalent. However, for a very special type, this work is quite easy.

qt0 qt1 qtk−1 CΛ := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ],

where Λ = {t0, ··· , tk−1}. Theorem (Csajb´ok,Marino, Polverino, Z, submitted)

Let Λ1 and Λ2 be two k-subsets of {0,..., n − 1}.

28/35 Then C1 and C2 are equivalent if and only if

Λ2 = Λ1 + s := {i + s (mod n): i ∈ Λ1}

for some s ∈ {0, ··· , n − 1}.

Moore matrices

In general, it is quite diﬃcult to tell whether C1 and C2 are equivalent. However, for a very special type, this work is quite easy.

qt0 qt1 qtk−1 CΛ := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ],

where Λ = {t0, ··· , tk−1}. Theorem (Csajb´ok,Marino, Polverino, Z, submitted)

Let Λ1 and Λ2 be two k-subsets of {0,..., n − 1}. Deﬁne n qi o C = P a X : a ∈ n for j = 1, 2. j i∈Λj i i Fq

28/35 if and only if

Λ2 = Λ1 + s := {i + s (mod n): i ∈ Λ1}

for some s ∈ {0, ··· , n − 1}.

Moore matrices

In general, it is quite diﬃcult to tell whether C1 and C2 are equivalent. However, for a very special type, this work is quite easy.

qt0 qt1 qtk−1 CΛ := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ],

where Λ = {t0, ··· , tk−1}. Theorem (Csajb´ok,Marino, Polverino, Z, submitted)

Let Λ1 and Λ2 be two k-subsets of {0,..., n − 1}. Deﬁne n qi o C = P a X : a ∈ n for j = 1, 2. Then C and C are j i∈Λj i i Fq 1 2 equivalent

28/35 Moore matrices

In general, it is quite diﬃcult to tell whether C1 and C2 are equivalent. However, for a very special type, this work is quite easy.

qt0 qt1 qtk−1 CΛ := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ],

where Λ = {t0, ··· , tk−1}. Theorem (Csajb´ok,Marino, Polverino, Z, submitted)

Let Λ1 and Λ2 be two k-subsets of {0,..., n − 1}. Deﬁne n qi o C = P a X : a ∈ n for j = 1, 2. Then C and C are j i∈Λj i i Fq 1 2 equivalent if and only if

Λ2 = Λ1 + s := {i + s (mod n): i ∈ Λ1}

for some s ∈ {0, ··· , n − 1}.

28/35 It is a q-analogue of Vandermonde matrices. Y det(MA) = (c0α0 + c1α1 + ··· ck−1αk−1), c

where c = (c0, c1, ··· , ck−1) runs over PG(k − 1, q). Therefore,

elements in A are Fq-linearly independent iﬀ det(MA) 6= 0.

Moore matrices

k For A := (α0, . . . , αk−1) ⊆ Fqn and k ≤ n, a square Moore matrix is deﬁned as   α0 α1 ··· αk−1  αq αq ··· αq   0 1 k−1  MA :=  . . . .  .  . . .. .    qk−1 qk−1 qk−1 α0 α1 ··· αk−1

29/35 Y det(MA) = (c0α0 + c1α1 + ··· ck−1αk−1), c

where c = (c0, c1, ··· , ck−1) runs over PG(k − 1, q). Therefore,

elements in A are Fq-linearly independent iﬀ det(MA) 6= 0.

Moore matrices

k For A := (α0, . . . , αk−1) ⊆ Fqn and k ≤ n, a square Moore matrix is deﬁned as   α0 α1 ··· αk−1  αq αq ··· αq   0 1 k−1  MA :=  . . . .  .  . . .. .    qk−1 qk−1 qk−1 α0 α1 ··· αk−1 It is a q-analogue of Vandermonde matrices.

29/35 Therefore,

elements in A are Fq-linearly independent iﬀ det(MA) 6= 0.

Moore matrices

where c = (c0, c1, ··· , ck−1) runs over PG(k − 1, q).

29/35 Moore matrices

where c = (c0, c1, ··· , ck−1) runs over PG(k − 1, q). Therefore,

elements in A are Fq-linearly independent iﬀ det(MA) 6= 0.

29/35 Besides I = {0, 1, ··· , k − 1}, it is interesting to ask whether there exist other I sharing the same property.

Elements in A are Fq-linearly independent iﬀ det(MA,I ) 6= 0.

We call such I a Moore exponent set for q and n.

Moore exponent sets

For any set of distinct nonnegative integers I = {t0, t1,..., tk−1} k and A = (α0, α1, . . . , αk−1) ⊆ Fqn , k ≤ n and let

 qt0 qt0 qt0  α0 α1 ··· αk−1  qt1 qt1 qt1   α0 α1 ··· αk−1  MA, I :=  . . . .  .  . . .. .   . . .  qtk−1 qtk−1 qtk−1 α0 α1 ··· αk−1

30/35 Elements in A are Fq-linearly independent iﬀ det(MA,I ) 6= 0.

We call such I a Moore exponent set for q and n.

Moore exponent sets

For any set of distinct nonnegative integers I = {t0, t1,..., tk−1} k and A = (α0, α1, . . . , αk−1) ⊆ Fqn , k ≤ n and let

 qt0 qt0 qt0  α0 α1 ··· αk−1  qt1 qt1 qt1   α0 α1 ··· αk−1  MA, I :=  . . . .  .  . . .. .   . . .  qtk−1 qtk−1 qtk−1 α0 α1 ··· αk−1 Besides I = {0, 1, ··· , k − 1}, it is interesting to ask whether there exist other I sharing the same property.

30/35 Moore exponent sets

For any set of distinct nonnegative integers I = {t0, t1,..., tk−1} k and A = (α0, α1, . . . , αk−1) ⊆ Fqn , k ≤ n and let

Elements in A are Fq-linearly independent iﬀ det(MA,I ) 6= 0.

We call such I a Moore exponent set for q and n.

30/35 Besides I = {0, 1, ··· , k − 1}, there are other known examples of Moore exponent sets. • I = {0, s, ··· , (k − 1)s} for any n satisfying gcd(s, n) = 1 (Generalized Gabidulin codes); • I = {0, 1, 3} for n = 7 with odd q (Csajb´ok,Marino, Polverino, Z.); • I = {0, 1, 3} for n = 8 with q ≡ 1 (mod 3) (Csajb´ok,Marino, Polverino, Z.).

A connection with MRD codes

Theorem

For q and n, I = {t0, ··· , tk−1} is a Moore exponent set if and only if

qt0 qt1 qtk−1 CI := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ]

is an MRD code.

31/35 • I = {0, s, ··· , (k − 1)s} for any n satisfying gcd(s, n) = 1 (Generalized Gabidulin codes); • I = {0, 1, 3} for n = 7 with odd q (Csajb´ok,Marino, Polverino, Z.); • I = {0, 1, 3} for n = 8 with q ≡ 1 (mod 3) (Csajb´ok,Marino, Polverino, Z.).

A connection with MRD codes

Theorem

For q and n, I = {t0, ··· , tk−1} is a Moore exponent set if and only if

qt0 qt1 qtk−1 CI := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ]

is an MRD code.

Besides I = {0, 1, ··· , k − 1}, there are other known examples of Moore exponent sets.

31/35 • I = {0, 1, 3} for n = 7 with odd q (Csajb´ok,Marino, Polverino, Z.); • I = {0, 1, 3} for n = 8 with q ≡ 1 (mod 3) (Csajb´ok,Marino, Polverino, Z.).

A connection with MRD codes

Theorem

For q and n, I = {t0, ··· , tk−1} is a Moore exponent set if and only if

qt0 qt1 qtk−1 CI := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ]

is an MRD code.

Besides I = {0, 1, ··· , k − 1}, there are other known examples of Moore exponent sets. • I = {0, s, ··· , (k − 1)s} for any n satisfying gcd(s, n) = 1 (Generalized Gabidulin codes);

31/35 • I = {0, 1, 3} for n = 8 with q ≡ 1 (mod 3) (Csajb´ok,Marino, Polverino, Z.).

A connection with MRD codes

Theorem

For q and n, I = {t0, ··· , tk−1} is a Moore exponent set if and only if

qt0 qt1 qtk−1 CI := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ]

is an MRD code.

31/35 A connection with MRD codes

Theorem

For q and n, I = {t0, ··· , tk−1} is a Moore exponent set if and only if

qt0 qt1 qtk−1 CI := {a0X + a1X + ... + ak−1X : ai ∈ Fqn } ⊆ Fqn [X ]

is an MRD code.

Besides I = {0, 1, ··· , k − 1}, there are other known examples of Moore exponent sets. • I = {0, s, ··· , (k − 1)s} for any n satisfying gcd(s, n) = 1 (Generalized Gabidulin codes); • I = {0, 1, 3} for n = 7 with odd q (Csajb´ok,Marino, Polverino, Z.); • I = {0, 1, 3} for n = 8 with q ≡ 1 (mod 3) (Csajb´ok,Marino, Polverino, Z.). 31/35 Proof: q even q q3 The Dickson matrix associated with X + X + X ∈ Fq7 [X ] is  1 1 0 1 0 0 0   0 1q 1q 0 1q 0 0     q2 q2 q2   0 0 1 1 0 1 0   3 3 3   0 0 0 1q 1q 0 1q   4 4 4   q q q  1 0 0 0 1 1 0   q5 q5 q5   0 1 0 0 0 1 1  1q6 0 1q6 0 0 0 1q6 Rank = 4 for q even ⇒ q3 roots ⇒ I is not an Moore exponent set.

n = 7

Theorem The set I = {0, 1, 3} is a Moore exponent set if and only if q is odd.

32/35 q q3 The Dickson matrix associated with X + X + X ∈ Fq7 [X ] is  1 1 0 1 0 0 0   0 1q 1q 0 1q 0 0     q2 q2 q2   0 0 1 1 0 1 0   3 3 3   0 0 0 1q 1q 0 1q   4 4 4   q q q  1 0 0 0 1 1 0   q5 q5 q5   0 1 0 0 0 1 1  1q6 0 1q6 0 0 0 1q6 Rank = 4 for q even ⇒ q3 roots ⇒ I is not an Moore exponent set.

n = 7

Theorem The set I = {0, 1, 3} is a Moore exponent set if and only if q is odd.

Proof: q even

32/35 Rank = 4 for q even ⇒ q3 roots ⇒ I is not an Moore exponent set.

n = 7

Theorem The set I = {0, 1, 3} is a Moore exponent set if and only if q is odd.

Proof: q even q q3 The Dickson matrix associated with X + X + X ∈ Fq7 [X ] is  1 1 0 1 0 0 0   0 1q 1q 0 1q 0 0     q2 q2 q2   0 0 1 1 0 1 0   3 3 3   0 0 0 1q 1q 0 1q   4 4 4   q q q  1 0 0 0 1 1 0   q5 q5 q5   0 1 0 0 0 1 1  1q6 0 1q6 0 0 0 1q6

32/35 Rank = 4 for q even ⇒ q3 roots ⇒ I is not an Moore exponent set.

n = 7

Theorem The set I = {0, 1, 3} is a Moore exponent set if and only if q is odd.

Proof: q even q q3 The Dickson matrix associated with X + X + X ∈ Fq7 [X ] is 1 1 0 1 0 0 0 0 1 1 0 1 0 0     0 0 1 1 0 1 0   0 0 0 1 1 0 1     1 0 0 0 1 1 0   0 1 0 0 0 1 1 1 0 1 0 0 0 1

32/35 n = 7

Theorem The set I = {0, 1, 3} is a Moore exponent set if and only if q is odd.

Proof: q even q q3 The Dickson matrix associated with X + X + X ∈ Fq7 [X ] is 1 1 0 1 0 0 0 0 1 1 0 1 0 0     0 0 1 1 0 1 0   0 0 0 1 1 0 1     1 0 0 0 1 1 0   0 1 0 0 0 1 1 1 0 1 0 0 0 1 Rank = 4 for q even ⇒ q3 roots ⇒ I is not an Moore exponent 32/35 set. • There exists u1, u2, u3 which are Fq-linearly independent.   u1 u2 u3 q q q  u u u  7 • The rows/columns of M :=  1 2 3  are Fq -linearly q3 q3 q3 u1 u2 u3 dependent.

• P := h(u , u , u )i , σ(x , x , x ) = (xq, xq, xq). 1 2 3 Fq7 1 2 3 1 2 3

• P, Pσ, Pσ3 are on a line `.

Proof: q odd

Suppose to the contrary that {0, 1, 3} is not a Moore exponent set.

33/35   u1 u2 u3 q q q  u u u  7 • The rows/columns of M :=  1 2 3  are Fq -linearly q3 q3 q3 u1 u2 u3 dependent.

• P := h(u , u , u )i , σ(x , x , x ) = (xq, xq, xq). 1 2 3 Fq7 1 2 3 1 2 3

• P, Pσ, Pσ3 are on a line `.

Proof: q odd

Suppose to the contrary that {0, 1, 3} is not a Moore exponent set.

• There exists u1, u2, u3 which are Fq-linearly independent.

33/35 • P := h(u , u , u )i , σ(x , x , x ) = (xq, xq, xq). 1 2 3 Fq7 1 2 3 1 2 3

• P, Pσ, Pσ3 are on a line `.

Proof: q odd

Suppose to the contrary that {0, 1, 3} is not a Moore exponent set.

• There exists u1, u2, u3 which are Fq-linearly independent.   u1 u2 u3 q q q  u u u  7 • The rows/columns of M :=  1 2 3  are Fq -linearly q3 q3 q3 u1 u2 u3 dependent.

33/35 • P, Pσ, Pσ3 are on a line `.

Proof: q odd

Suppose to the contrary that {0, 1, 3} is not a Moore exponent set.

• There exists u1, u2, u3 which are Fq-linearly independent.   u1 u2 u3 q q q  u u u  7 • The rows/columns of M :=  1 2 3  are Fq -linearly q3 q3 q3 u1 u2 u3 dependent.

• P := h(u , u , u )i , σ(x , x , x ) = (xq, xq, xq). 1 2 3 Fq7 1 2 3 1 2 3

33/35 Proof: q odd

Suppose to the contrary that {0, 1, 3} is not a Moore exponent set.

• There exists u1, u2, u3 which are Fq-linearly independent.   u1 u2 u3 q q q  u u u  7 • The rows/columns of M :=  1 2 3  are Fq -linearly q3 q3 q3 u1 u2 u3 dependent.

• P := h(u , u , u )i , σ(x , x , x ) = (xq, xq, xq). 1 2 3 Fq7 1 2 3 1 2 3

• P, Pσ, Pσ3 are on a line `.

33/35 • ` 6= `σ, the length of the orbit of ` under σ is 7.

• Pσi , `σi for i = 0,..., 6 form a Fano plane.

• A Fano plane cannot be embedded in PG(2, q7) with q odd.

• A contradiction!

q q3 3 • Therefore, α1X + α2X + α3X cannot have q roots.

Proof: q odd

• σ(x , x , x ) = (xq, xq, xq), ` = hP, Pσ, Pσ3 i . 1 2 3 1 2 3 Fq7

34/35 • Pσi , `σi for i = 0,..., 6 form a Fano plane.

• A Fano plane cannot be embedded in PG(2, q7) with q odd.

• A contradiction!

q q3 3 • Therefore, α1X + α2X + α3X cannot have q roots.

Proof: q odd

• σ(x , x , x ) = (xq, xq, xq), ` = hP, Pσ, Pσ3 i . 1 2 3 1 2 3 Fq7

• ` 6= `σ, the length of the orbit of ` under σ is 7.

34/35 • A Fano plane cannot be embedded in PG(2, q7) with q odd.

• A contradiction!

q q3 3 • Therefore, α1X + α2X + α3X cannot have q roots.

Proof: q odd

• σ(x , x , x ) = (xq, xq, xq), ` = hP, Pσ, Pσ3 i . 1 2 3 1 2 3 Fq7

• ` 6= `σ, the length of the orbit of ` under σ is 7.

• Pσi , `σi for i = 0,..., 6 form a Fano plane.

34/35 • A contradiction!

q q3 3 • Therefore, α1X + α2X + α3X cannot have q roots.

Proof: q odd

• σ(x , x , x ) = (xq, xq, xq), ` = hP, Pσ, Pσ3 i . 1 2 3 1 2 3 Fq7

• ` 6= `σ, the length of the orbit of ` under σ is 7.

• Pσi , `σi for i = 0,..., 6 form a Fano plane.

• A Fano plane cannot be embedded in PG(2, q7) with q odd.

34/35 q q3 3 • Therefore, α1X + α2X + α3X cannot have q roots.

Proof: q odd

• σ(x , x , x ) = (xq, xq, xq), ` = hP, Pσ, Pσ3 i . 1 2 3 1 2 3 Fq7

• ` 6= `σ, the length of the orbit of ` under σ is 7.

• Pσi , `σi for i = 0,..., 6 form a Fano plane.

• A Fano plane cannot be embedded in PG(2, q7) with q odd.

• A contradiction!

34/35 Proof: q odd

• σ(x , x , x ) = (xq, xq, xq), ` = hP, Pσ, Pσ3 i . 1 2 3 1 2 3 Fq7

• ` 6= `σ, the length of the orbit of ` under σ is 7.

• Pσi , `σi for i = 0,..., 6 form a Fano plane.

• A Fano plane cannot be embedded in PG(2, q7) with q odd.

• A contradiction!

q q3 3 • Therefore, α1X + α2X + α3X cannot have q roots.

34/35 Theorem (Bartoli, Z. Submitted) Assume that I is not an arithmetic progression. Then there exist integers N and Q ≤ 5 depending on I such that I is not a Moore exponent set over Fqn provided that q > Q and n > N.

We believe that the restriction on q > Q can be removed.

An asymptotic classiﬁcation result

In general it seems illusive to give a complete list of Moore exponent set, because the associated Dickson matrices are getting larger.

35/35 We believe that the restriction on q > Q can be removed.

An asymptotic classiﬁcation result

In general it seems illusive to give a complete list of Moore exponent set, because the associated Dickson matrices are getting larger.

Theorem (Bartoli, Z. Submitted) Assume that I is not an arithmetic progression. Then there exist integers N and Q ≤ 5 depending on I such that I is not a Moore exponent set over Fqn provided that q > Q and n > N.

35/35 An asymptotic classiﬁcation result

In general it seems illusive to give a complete list of Moore exponent set, because the associated Dickson matrices are getting larger.

We believe that the restriction on q > Q can be removed.

35/35 An asymptotic classiﬁcation result

In general it seems illusive to give a complete list of Moore exponent set, because the associated Dickson matrices are getting larger.

Conjecture Assume that I is not an arithmetic progression. Then there exist integers N /////and////////Q ≤ 5 depending on I such that I is not a Moore exponent set over Fqn provided that ////////q > Q/////and n > N.

We believe that the restriction on q > Q can be removed.

35/35 Thanks for your attention!

35/35