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There is no general classification of MUBs. The main open problem in this area is to construct a maximal number of MUBs for any given n. It is known that the maximal set of MUBs of Cn consists of at most n + 1 bases, and sets attaining this bound are called complete sets of MUBs. Constructions of complete sets are known only for prime power dimensions. Even for the smallest non-prime power dimension six the problem of finding a maximal set of MUBs is extremely hard [37] and remains open after more than 30 years. Several kinds of constructions of MUBs are available [5, 8, 24, 26, 34, 35, 48]. As a matter of fact, essentially there are only three types of constructions up to now: constructions associated with symplectic spreads, using planar functions over fields of odd characteristic, and Gow’s construction in [24] of a unitary matrix (although it seems it is isomorphic to the classical one). On the other hand one can consider MUBs in the Euclidean space Rn (real MUBs), and in this case the upper bound for a maximal set is n/2 + 1. Constructions of real MUBs are strictly connected to algebraic coding theory (optimal Kerdock type codes) and the notion of extremal line-sets in Euclidean spaces [4, 13]. LeCompte et al. [38] characterized collections of real MUBs in terms of association schemes. In [4] connections of real MUBs with binary and quaternary Kerdock and Preparata codes [25, 39], and association schemes were studied. We also note that one can come to Kerdock and Preparata codes through integral lattices associated with orthogonal decompositions of Lie algebras sln(C) [2, 3]. In this paper we give explicit descriptions of complete sets of MUBs and orthogonal decom- positions of special Lie algebras sln(C) obtained from commutative and symplectic semifields, and from some other non-semifield symplectic spreads. We provide direct formulas to construct MUBs from semifields. We show that automorphism groups of complete sets of MUBs and corresponding orthogonal decompositions of Lie algebras sln(C) are isomorphic, and in the case of symplectic spreads these automorphism groups are determined by the automorphism groups of those spreads. Automorphism groups are important invariants, therefore they can be used to show inequivalence of the various known types of MUBs. Planar functions over fields of odd characteristics also lead to constructions of MUBs. There are no planar functions over fields of characteristic two. However, Zhou [51] proposed a new notion of “planar” functions over fields of characteristic two. Based on this notion, we propose new constructions of MUBs (but we prefer to call these functions pseudo-planar). We also propose a generalization of the notion of pseudo-planar functions for arbitrary characteristic. MUBs can be constructed using different objects. We study their mutual relations. Connec- tions between them are given in the following road map:

2 Complete ✛ ✲Orth. Decom. of C set of MUBs Lie alg. sln( ) ✻ ✻

Planar function Symplectic (Pseudo-planar) Spread ✻ ✻

Commutative ✛ ✲ Symplectic Presemifield Presemifield

This map shows that starting from a commutative presemifield one can construct consec- utively (pseudo-)planar functions and then MUBs. On the other hand, one can start from a commutative presemifield and then construct consecutively symplectic presemifield, symplectic spread, orthogonal decomposition of Lie algebra and finally MUBs. We will show that in fact our diagram is “commutative”: we will not get new MUBs moving from one construction to others. Note that there are planar functions not coming from presemifields, symplectic spreads not related to semifields [7, 27, 28, 29, 33, 46] and orthogonal decompositions not related to symplectic spreads.

2 Automorphism groups

The Lie algebra L = sl (C) is the algebra of n n traceless matrices over C, where the operation n × of multiplication is given by the commutator of matrices: [A, B]= AB BA. A subalgebra H − of a Lie algebra L is called a Cartan subalgebra if it is nilpotent and equal to its normalizer, which is the set of those elements X in L such that [X,H] H. In case of L = sl (C) Cartan ⊆ n subalgebras are maximal abelian subalgebras and they are conjugate under automorphisms of the Lie algebra. In particular, they are all conjugate to the standard Cartan subalgebra, consisting of all traceless diagonal matrices. A decomposition of a simple Lie algebra L into a direct sum of Cartan subalgebras L = H H H 0 ⊕ 1 ⊕···⊕ n is called an orthogonal decomposition [37], if the subalgebras Hi are pairwise orthogonal with respect to the Killing form K(A, B) on L. Recall that the Killing form on a Lie algebra L is defined by K(A, B) = Tr(adA adB), where the operator adA : L L is given by adA (C) = · → [A, C] and Tr is the trace. The Killing form is symmetric and non-degenerate on L. In the case of L = sln(C) we have K(A, B) = 2nTr(AB). t The adjoint operation on the set of n n complex matrices is given by A = A (conjugate ∗ × ∗ transpose). A Cartan subalgebra is called closed under the adjoint operation if H∗ = H.

Theorem 2.1 ([11], Theorem 5.2) Complete sets of MUBs in Cn are in one-to-one corre- spondence with orthogonal decompositions of Lie algebra sln(C) such that all Cartan subalgebras in this decomposition are closed under the adjoint operation.

3 This correspondence is given in the following way. If = B ,B ,...,B is a complete B { 0 1 n} set of MUBs then the associated Cartan subalgebra Hi consists of all traceless matrices that are diagonal with respect to the basis Bi (in other words, vectors of the basis Bi are common eigenvectors of all matrices in Hi). The automorphism group Aut( ) [37] of an orthogonal decomposition of a Lie algebra L D D consists of all automorphisms of L preserving : D Aut( )= ϕ Aut(L) ( i)( j) ϕ(H )= H . D { ∈ | ∀ ∃ i j} Recall that Aut(L) = Inn(L) T , where Inn(L) = PSL (C) is the group of all inner · h i ∼ n automorphisms of L and T is the outer automorphism A At. 7→ − Let = B ,B ,...,B be a complete set of MUBs of Cn. With any orthonormal basis B { 0 1 n} Bi we can associate an orthoframe Ort(Bi) of 1-spaces generated by vectors of Bi. We note that if we consider the representations of all bases Bi in some fixed standard basis, then the conjugation map τ(x1,...,xn)= x = (x1,..., xn) sends one orthoframe Ort(Bi) to other orthoframe. Define

P GL (C)+ = P GL (C) τ , n n h i Aut( )= ψ P GL (C)+ ( i)( j) ψ(Ort(B )) = Ort(B ) . B { ∈ n | ∀ ∃ i j }

Theorem 2.2 Let be an orthogonal decomposition of the Lie algebra sln(C) and be the corresponding completeD set of MUBs. Then B

Aut( ) = Aut( ). D ∼ B Proof. Let ϕ Aut( ). For any automorphism ϕ of the Lie algebra sl (C) we have ϕ = ϕ ∈ D n X or ϕ = ϕX T , where 1 ϕX (A)= XAX− for some matrix X SL (C), and ∈ n T (A)= At. − Let H be a Cartan subalgebra from and B = e ,...,e be the corresponding basis, so for D { 1 n} any h H we have he = α (h)e for some linear map α : H C. Assume that ϕ = ϕ . Then ∈ i i i i → X the vectors fi = Xei generate the orthoframe associated with the Cartan subalgebra ϕ(H):

1 1 XhX− fi = XhX− Xei = Xαi(h)ei = αi(h)fi.

Therefore, the matrix X generates an element from Aut( ). t B Let ϕ = ϕ T . Note that for h H one has h = h H by Theorem 2.1. Then the vectors X ∈ ∗ ∈ fi = Xei determine a basis corresponding to Cartan subalgebra ϕ(H):

t 1 1 1 (ϕ T h)f = Xh X− f = Xh X− f = Xh X− Xe = Xh e = X i − i − ∗ i − ∗ i − ∗ i Xα (h )e = Xα (h )e = α (h )Xe = α (h )f . − i ∗ i − i ∗ i − i ∗ i − i ∗ i Therefore, Xτ generates an element from Aut( ). B Conversely, let ψ Aut( ). Suppose first that ψ is generated by the matrix X GL (C). ∈ B ∈ n We can assume that det X = 1 (otherwise we multiply X by an appropriate scalar matrix).

4 Let B = e be a basis from and let ψ map the orthoframe Ort(B) to another orthoframe { i} B Ort(B ). Denote f = Xe . Let H = ϕ (H)= XHX 1 and h H. Then ′ i i ′ X − ∈ 1 ϕX (h)fi = XhX− Xei = Xhei = Xαi(h)ei = αi(h)Xei = αi(h)fi.

Therefore, the vectors fi are common eigenvectors for H′, so H′ is the Cartan subalgebra associated with the basis B . Hence ϕ Aut( ). ′ X ∈ D Now assume that ψ is generated by Xτ. Then the vectors fi = Xei from an orthoframe B′ are common eigenvectors of matrices of the Cartan subalgebra ϕX T (H): t 1 t (ϕ T h)f = Xh X− Xe = Xh e = Xh e = Xh e = X i − i − i − ∗ i − ∗ i Xα (h )e = Xα (h )e = α (h )Xe = α (h )f , − i ∗ i − i ∗ i − i ∗ i − i ∗ i so ϕ T Aut( ).  X ∈ D All known constructions of complete sets of MUBs were obtained with the help of symplectic spreads and planar functions, and the construction from [24]. Now we recall constructions of orthogonal decompositions of Lie algebras sln(C) associated with symplectic spreads [37, 30]. Let F = Fq be a finite field of order q. Let V be a vector space over F with the usual dot product u v. We can consider W = V V as a vector space over the prime field F and define · ⊕ p an alternating bilinear form on W by

(u, v), (u′, v′) = tr(u v′ v u′), (1) h i · − · where tr is a trace function from Fq to Fp. Let n = V and let e denote the standard basis of Cn, indexed by elements of V . Let | | { w} ε C be a primitive pth root of unity. For u V , the generalized Pauli matrices are defined as ∈ ∈ the following n n matrices: × X(u) : ew eu+w 7→ tr(v w) Z(v) : e ε · e w 7→ w The matrices Du,v = X(u)Z(v) form a basis of the space of complex square matrices of size n n. Moreover, the matrices D , × u,v (u, v) = (0, 0), generate the Lie algebra sl (C). Note that 6 n tr(v u′) (u,v),(u′ ,v′) [D , D ′ ′ ]= ε · (1 εh i)D ′ ′ , u,v u ,v − u+u ,v+v so [Du,v, Du′,v′ ] = 0 if and only if (u, v), (u′, v′) = 0. h i r Let V be an r-dimensional space over Fp (so n = p ). A symplectic spread of the symplectic 2r-dimensional space W = V V over F is a family of n + 1 totally isotropic r-subspaces of ⊕ p W such that every nonzero point of W lies in a unique subspace. Thus such r-subspaces are maximal totally isotropic subspaces. Let Σ = W ,W ,...,W be a symplectic spread. Then { 0 1 n} L = H H H , 0 ⊕ 1 ⊕···⊕ n H = D (u, v) W , i h u,v | ∈ ii gives [37, 30] the corresponding orthogonal decomposition of the Lie algebra sln(C). We define

Sp±(W )= ϕ GL(W ) ( s = 1)( u, v W ) ϕ(u), ϕ(v) = s u, v { ∈ | ∃ ± ∀ ∈ h i h i} Aut(Σ) = ϕ Sp±(W ) ( i)( j) ϕ(W )= W { ∈ | ∀ ∃ i j}

5 Theorem 2.3 ([37], Proposition 1.3.3.) Let be the orthogonal decomposition of the Lie D algebra sln(C) corresponding to a symplectic spread Σ. Then

Aut( )= K.Aut(Σ), D where K is the set of all automorphisms that fix every line D . h u,vi In the case of odd characteristic, K is isomorphic to W = V 2 and an embedding of K in PSLn(C) is given through conjugations by matrices Du,v:

1 (u,v),(a,b) Du,vDa,bDu,v− = ε−h iDa,b.

The outer automorphism T is given by:

tr(a b) T (Da,b)= ε− · D a,b, − − 2 Z therefore in the case of even characteristic, we have K ∼= V . 2. Any symplectic spread Σ determines a translation plane A(Σ). The collineation group of A(Σ) is a semidirect product of the translation group V 2 and the translation complement. However, the group extension in Theorem 2.3 can be nonsplit (see Section 4.1).

3 Symplectic spreads in odd characteristics and MUBs

In this section we show how to construct a complete set of MUBs directly from symplectic spreads and semifields. Throughout this section F = Fq denotes a finite field of odd order q = pr. Let ω C be a primitive pth root of unity. ∈ Lemma 3.1 Let Σ be a symplectic spread of W = V V , and let h : V V be a mapping such ⊕ → that (u, h(u)) u V is a maximal totally isotropic subspace. Then tr(u h(w)) = tr(h(u) w) { | ∈ } · · for all u V , w V . ∈ ∈ Proof. tr(u h(w) h(u) w)= ((u, h(u)), (w, h(w)) = 0.  · − · h i Lemma 3.2 Let Σ be a symplectic spread of W = V V , and let h : V V be a linear mapping ⊕ → such that (u, h(u)) u V is a maximal totally isotropic subspace. Then for any v V , the vector { | ∈ } ∈ tr( 1 w h(w)+v w) bh,v = ω 2 · · ew w V X∈ is an eigenvector of D for all u V . u,h(u) ∈ Proof. Indeed,

tr( 1 w h(w)+v w+h(u) w) Du,h(u)(bh,v) = ω 2 · · · ew+u w V X∈ tr( 1 (w u) h(w u)+v (w u)+h(u) (w u)) = ω 2 − · − · − · − ew w V X∈ tr( 1 w h(w)+v w)+ 1 tr(h(u) w u h(w)) tr( 1 u h(u)+v u) = ω 2 · · 2 · − · − 2 · · ew w V X∈ tr( 1 u h(u)+v u) = ω− 2 · · bh,v

6 by Lemma 3.1.  Lemma 3.2 allows us to construct the complete set of MUBs corresponding to the orthogonal decomposition of the Lie algebra sln(C) obtained from a symplectic spread. Symplectic spreads can be constructed using semifields. A finite presemifield is a ring with no zero-divisors, and with left and right distributivity [18]. A presemifield with multiplicative identity is called a semifield. A finite presemifield can be obtained from a finite field (F, +, ) · by introducing a new product operation , so it is denoted by (F, +, ). Every presemifield ∗ ∗ determines a spread Σ consisting of subspaces (0, F ) and (x,x y) x F , y F . A { ∗ | ∈ } ∈ presemifield is called symplectic if the corresponding spread is symplectic with respect to some alternating form [31] (so the spread might not be symplectic with respect to other forms). Two presemifields (F, +, ) and (F, +, ⋆) are called isotopic if there exist three bijective linear ∗ mappings L, M, N : F F such that → L(x y)= M(x) ⋆ N(y) ∗ for any x,y F . If M = N then the presemifields are called strongly isotopic. Every presemi- ∈ field is isotopic to a semifield. Isotopic semifields determine isomorphic planes. Lemma 3.2 implies

Theorem 3.3 Let (F, +, ) be a finite symplectic presemifield of odd characteristic. Then the following set forms a complete◦ set of MUBs:

B = ew w F , Bm = bm,v v F , m F, ∞ { | ∈ } { | ∈ } ∈

1 tr( 1 w (w m)+v w) bm,v = ω 2 · ◦ · ew. √q w F X∈ A function f : F F is called planar if → x f(x + a) f(x), (2) 7→ − is a permutation of F for each a F . Any planar function over F allows us to construct a ∈ ∗ complete set of MUBs.

Theorem 3.4 ([21, 34, 41]) Let F be a finite field of odd order q and f be a planar function. Then the following forms a complete set of MUBs:

B = ew w F , Bm = bm,v v F , m F, ∞ { | ∈ } { | ∈ } ∈

1 tr( 1 mf(w)+vw) bm,v = ω 2 ew. √q w F X∈ Every commutative presemifield of odd order corresponds to a quadratic planar polynomial, and vice versa. If (F, +, ) is a commutative presemifield then f(x)= x x is a planar function, ∗ ∗ and one can use Theorem 3.4 for constructing MUBs. Up to now, there is only one known type of planar function [15], which is not related to semifields:

k f(x)= x(3 +1)/2, where q = 3r, gcd(k, 2r) = 1, k 1 (mod 2r). 6≡ ± On the other hand, from a commutative presemifield one can construct a symplectic presemi- field (using Knuth’s cubical array method [36]) and then construct consecutively a symplectic

7 spread, an orthogonal decomposition of the Lie algebra sln(C) and finally MUBs. Below we show that we will not get new MUBs by these operations (a similar statement is true for even characteristic, see Section 4). pi+pj Let f(x)= i j aijx be a quadratic planar polynomial. Then the corresponding com- mutative presemifield≤ (F, +, ) is given by: P ∗ 1 1 i j j i x y = (f(x + y) f(x) f(y)) = a (xp yp + xp yp ). ∗ 2 − − 2 ij i j X≤ It defines a spread Σ consisting of subspaces (0, F ) and (x,x y) x F , y F . Starting { ∗ | ∈ } ∈ from Σ we will construct a symplectic spread using the Knuth cubical arrays method ([31], Proposition 3.8). The dual spread Σd is a spread of the dual space of F F . We can identify ⊕ that dual space with F F by using the alternating form (1). For each y F we have to find ⊕ ∈ all (u, v) such that (x,x y), (u, v) = 0 for all x F . We have h ∗ i ∈ (x,x y), (u, v) = tr(xv u(x y)) h ∗ i − ∗ 1 i j j i = tr(xv u a (xp yp + xp yp )) − 2 ij i j X≤ 1 r−i r−i j−i 1 r−j r−j r+i−j = tr(xv ap up xyp ap up xyp ) − 2 ij − 2 ij i j i j X≤ X≤ 1 r−i r−i j−i 1 r−j r−j r+i−j = tr(x(v ap up yp ap up yp )), − 2 ij − 2 ij i j i j X≤ X≤ which implies 1 r−i r−i j−i 1 r−j r−j r+i−j v = ap up yp + ap up yp . 2 ij 2 ij i j i j X≤ X≤ Therefore, Σd corresponds to the presemifield (F, +, ) defined by • 1 r−i r−i j−i 1 r−j r−j r+i−j u y = ap up yp + ap up yp . • 2 ij 2 ij i j i j X≤ X≤ Then the presemifield (F, +, ) with multiplication ◦ 1 r−i j−i r−i 1 r−j r+i−j r−j x y = y x = ap xp yp + ap xp yp ◦ • 2 ij 2 ij i j i j X≤ X≤ d d defines a symplectic spread Σ ∗. It is straightforward to check directly that the spread Σ ∗ with subspaces (0, F ) and (x,x y) x F , y F , is symplectic with respect to the form (1). { ◦ | ∈ } ∈ Then by Theorem 3.3 the following bases will form a complete set of MUBs:

B = ew w F , Bm = bm,v v F , m F, ∞ { | ∈ } { | ∈ } ∈

1 tr( 1 m(w w)+vw) bm,v = ω 2 ∗ ew, √q w F X∈ since

1 1 r−i j−i r−i 1 r−j r+i−j r−j tr w (w m) = tr wap wp mp + wap wp mp 2 · ◦ 4 ij 4 ij    i j i j X≤ X≤   8 1 i j 1 j i = tr wp a wp m + wp a wp m 4 ij 4 ij  i j i j X≤ X≤ 1  = tr m(w w) . 2 ∗   Remark 1. It is easy to see that strongly isotopic commutative semifields provide equivalent + MUBs (equivalence under the action of the group P GLn(C) ). Theorems 2.2 and 2.3 can be used to show inequivalence of MUBs obtained from nonisotopic semifields. Practically they distinguish all known types of MUBs.

3.1 Desarguesian spreads Desarguesian spreads are constructed with the help of finite fields, and the corresponding com- mutative and symplectic semifields are the same: x y = x y = xy, and the planar function is ∗ ◦ f(x)= x2. Godsil and Roy [23] showed that the constructions of Alltop [5], Ivanovi´c[26], Wooters and Fields [48], Klappenecker and R¨otteler [35], and Bandyopadhyay, Boykin, Roychowdhury and Vatan [8] are all equivalent to particular cases of this construction. It seems that Gow’s construction is equivalent to this case as well. Theorem 2.2 and [37] imply that for the automorphism group of the corresponding complete set of MUBs we have Aut( ) = F 2.(SL (q).Z ).Z , B ∼ 2 r 2 where extensions are split.

3.2 Spreads from Albert’s generalized twisted fields Let ρ be a nontrivial automorphism of a finite field F such that 1 F ρ 1. It means that F has − 6∈ − odd degree over the fixed field Fρ of ρ. Let V = F . Then the BKLA [6] symplectic spread Σ of −1 W = V V consists of the subspace (0,y) y F and subspaces (x,mxρ +mρxρ) x F , ⊕ { | ∈ } { −1 | ∈ } m F . This spread arises from a presemifield given by the operation x m = mxρ + mρxρ. ∈ k ◦ If we start from the planar function f(x)= xp +1 = xρ+1, then the commutative presemifield and corresponding symplectic presemifield are given by 1 x y = (xρy + xyρ), ∗ 2

1 −1 −1 x y = (xρy + xρ yρ ). ◦ 2

3.3 Ball-Bamberg-Laurauw-Penttila symplectic spread This non-semifield spread [7] is obtained from the previous one with the help of a technique known as “net replacement”. Let Σ be the BKLA spread. One can change this spread and get a new spread Σ in the following way. For all s F the map σ (v, w) = (sv,s 1w) is an ′ ∈ ∗ s − isometry of V V with respect to the form , . Denote the group of all σ by G. Let τ be an ⊕ h i s involution of V V which switches coordinates: (v, w) (w, v). Then Σ is G-invariant. For ⊕−1 7→ W = (x,xρ + xρ ) x F we consider the G-orbit 1 { | ∈ }

= σ (W ) s F ∗ . N { s 1 | ∈ }

9 Note that and τ( ) are G-invariant. Now we define N N

′ = τ( ). N N Then Σ′ = (Σ ) ′ \N ∪N is a symplectic spread with respect to the form (1).

Lemma 3.5 Under the action of the group G the spread Σ has the following four orbits: (x, 0) −1 { | x F ; (0,y) y F ; subspaces (x,mxρ + mρxρ) x F with nonsquare elements ∈ } { | ∈ } −1 { | ∈ } m F ; and subspaces (x,mxρ + mρxρ) x F with square elements m F . ∈ ∗ { | ∈ } ∈ ∗ Proof. We have

ρ−1 ρ ρ 1 ρ−1 1 ρ ρ σs(x,mx + m x ) = (sx,s− mx + s− m x ) 1 ρ−1 ρ−1 1 ρ ρ ρ = (sx,s− − m(sx) + s− − m (sx) ) 1 ρ−1 ρ−1 1 ρ ρ ρ = (u, s− − mu + s− − m u ), where u = sx. Therefore, the isometry σs sends the subspace indexed by m to the subspace 1 ρ−1 1 ρ−1 ρ+1 ρ−1 indexed by s− − m. Since s− − = (s )− , it remains for us to show that the set sρ+1 s F is the set of all squares in F . Indeed, assume that the fixed field F of ρ { | ∈ ∗} ∗ ρ has q elements. Then ρ = qk, F = qt , gcd(t, k) = 1, and t is an odd integer. We have 1 1 | | 1 gcd(qk + 1,qt 1) = 2, so sρ+1 s F is the set of all squares in F .  1 1 − { | ∈ ∗} ∗ −1 Lemma 3.6 Suppose the function α : F F is given by α(u) = uρ + uρ. Let β = α 1 and → − let the order of ρ be t. Then

1 ρ ρt−1 β(v)= (a0v + a1v + + at 1v ), 2 · · · − where a = a = 1, a = a = 1, i 0, for t 1 (mod 4), and a = a = 1, 4i 4i+1 4i+2 4i+3 − ≥ ≡ 4i 4i+3 − a = a = 1, i 0, for t 3 (mod 4). 4i+1 4i+2 ≥ ≡ Proof. We have

ρ2 1 ρt−1 β(v)+ β(v) = (a0v + + at 1v ) 2 · · · − 1 ρ2 ρt−1 ρ + (a0v + + at 3v + at 2v + at 1v ) 2 · · · − − − t 1 1 ρ − ρk ρ = ((a0 + at 2)v + (a1 + at 1)v + (ak + ak 2)v )= v . 2 − − − Xk=2 −1 Therefore β(v)ρ + β(v)ρ = v.  We use this function β for the following

Corollary 3.7 The spread Σ′ consists of the following subspaces: (x, 0) x F ; (0,y) y −1 { | ∈ } { | ∈ F ; subspaces (x,mxρ + mρxρ) x F with nonsquare elements m F ; and subspaces } { | ∈ } ∈ (x,sβ(xs)) x F with s F / 1 . { | ∈ } ∈ ∗ h− i

10 1 ρ−1 ρ Proof. The collection ′ consists of elements of the form (s− (u + u ), su). We denote 1 ρ−1 ρ N ρ−1 ρ s− (u + u ) = x. Then u + u = xs, u = β(xs) and su = sβ(xs). Therefore the corresponding spread subspaces are (x,sβ(xs)) x F , s F . Finally we note that s and { | ∈ } ∈ ∗ s determine the same subspaces.  − From this corollary we get the following complete set of MUBs:

1 tr(vw) B = ew w F , B0 = ω ew v F , ∞ { | ∈ } {√q | ∈ } w F X∈ 1 tr(mρwρ+1+vw) Bm = ω ew v F , m F ∗, m is nonsquare, {√q | ∈ } ∈ w F X∈ 1 tr( 1 wsβ(ws)+vw) Bs = ω 2 ew v F , s F ∗/ 1 . {√q | ∈ } ∈ h− i w F X∈ We note that the Ball-Bamberg-Laurauw-Penttila symplectic spread is not a semifield spread. If F is a field of order 27, then the corresponding plane is the Hering plane and its translation complement is isomorphic to SL (13). If F = q3 and F = q then the corresponding plane 2 | | 1 | ρ| 1 is the Suetake [44] plane and the translation component has order 3(q 1)(q3 1). 1 − 1 − 3.4 Other known semifields In this section we consider examples of known commutative and symplectic semifields. They are presented in pairs, so that symplectic semifields produce symplectic spreads with respect to the alternating form (1). The Dickson commutative semifield [18, 19] and corresponding Knuth [36] symplectic pre- semifields are defined by (a, b) (c, d) = (ac + jbσdσ, ad + bc), ∗ −1 −1 (a, b) (c, d) = (ac + bd, ad + jσ bcσ ), ◦ where q is odd, j be a nonsquare element of F = F , 1 = σ Aut(F ) and V = F F . q 6 ∈ ⊕ The Cohen-Ganley commutative semifield [14] and the corresponding symplectic Thas-Payne presemifield [45] are defined by

(a, b) (c, d) = (ac + jbd + j3(bd)9, ad + bc + j(bd)3), ∗ (a, b) (c, d) = (ac + bd, ad + jbc + j1/3bc1/9 + j1/3bd1/3), ◦ where q = 3r 9 and j F is nonsquare. ≥ ∈ Ganley commutative semifields [22] and corresponding symplectic presemifields are defined by (a, b) (c, d) = (ac b9d bd9, ad + bc + b3d3), ∗ − − (a, b) (c, d) = (ac + bd, ad + bd1/3 b1/9c1/9 b9c), ◦ − − t where F = Fq, q = 3 , and t 3 odd . ≥ 10 The Penttila-Williams sporadic symplectic semifield [40] (F 5 F 5 , +, ) of order 3 and 3 ⊕ 3 ◦ the corresponding commutative semifield are given by

(a, b) (c, d) = (ac + bd, ad + bd9 + bc27), ◦ (a, b) (c, d) = (ac + (bd)9, ad + bc + (bd)27). ∗ During last several years some families of quadratic planar functions were discovered [10, 12, 15, 16, 17, 20, 47, 49, 50, 52].

11 4 Symplectic spreads in even characteristics and MUBs

In this section we use symplectic spreads in even characteristic for constructing MUBs. Con- structions of MUBs are also given in [34], where related objects are considered over prime fields. We will use Galois rings, so descriptions can be given explicitly (especially in the case of pre- r semifields). Throughout this section F = Fq denotes a finite field of even order q = 2 . Let ω C be a primitive 4th root of unity. ∈ To write corresponding MUBs, we need the Galois ring R = GR(4r) of characteristic 4 and r F cardinality 4 . We recall some facts about R [25, 39]. We have R/2R ∼= q, the unit group R = R 2R contains a cyclic subgroup C of size 2r 1 isomorphic to F . The set = 0 C ∗ \ − q∗ T { }∪ is called the Teichm¨uller set in R. Every element x R can be written uniquely in the form ∈ x = a + 2b for a, b . Then ∈T r−1 r−1 Tr(x) = (a + a2 + + a2 ) + 2(b + b2 + + b2 ). · · · · · · Since R/2R = F , for every element u F there exists a corresponding unique elementu ˆ , ∼ q ∈ q ∈T called the Teichm¨uller lift of u. If x, y, z and z x + y (mod 2R) then ∈T ≡ z = x + y + 2√xy. The last equation can be written in the form wˆ =u ˆ +v ˆ + 2√uˆvˆ for elements w, u, v F , w = u + v. ∈ q Let (F, +, ) be a presemifield with multiplication ◦ i j x y = a x2 y2 . ◦ ij Then we extend this multiplication to Xin the following way: T ×T i j xˆ yˆ = a xˆ2 yˆ2 ◦ ij Lemma 4.1 Let (F, +, ) be a symplectic presemifield.X Then ◦ c 1. tr(x (z y)) = tr(z (x y)) for all x, y, z F . · ◦ · ◦ ∈ 2. Tr(ˆx (ˆz yˆ)) = Tr(ˆz (ˆx yˆ)) for all x, y, z F . · ◦ · ◦ ∈ Proof. 1. tr(x (z y) z (x y)) = ((x,x y), (z, z y) = 0. · ◦ − · ◦ h ◦ ◦ i 2. We have i j i j tr(z (x y) x (z y)) = tr(z a x2 y2 x a z2 y2 ) · ◦ − · ◦ ij − ij i j −i −i j−i = tr(z X a x2 y2 Xa2 x2 zy2 ) ij − ij i j −i −i j−i = tr(z(X a x2 y2 X a2 x2 y2 )) = 0, ij − ij where powers of 2 and indices are considered moduloX r for convenience.X Therefore, 2i 2j 2−i 2−i 2j−i 2i 2i 2j+i 2i 2i 2j aijx y = aij x y = ar i,jx y = ar i,j ix y . − − − i X 2i X \ 2 X X Hence aij = ar i,j i and aij = ar i,j i . Then − − − − i j i j Tr(ˆz (ˆx yˆ) xˆ (ˆz yˆ)) = Tr(ˆz a xˆ2 yˆ2 xˆ a zˆ2 yˆ2 ) · ◦ − · ◦c ij − ij i j −i −i j−i = Tr(ˆz(X a xˆ2 yˆ2 Xa 2 xˆ2 yˆ2 )) cij − ijc X 2i 2j X 2i 2i 2j+i = Tr(ˆz( aijxˆ yˆ a\r i,j xˆ yˆ )) c − c− X 2i 2j X 2i 2i 2j = Tr(ˆz( aijxˆ yˆ a\r i,j i xˆ yˆ )) = 0.  c − − − X X c 12 Theorem 4.2 Let (F, +, ) be a symplectic presemifield. Then the following forms a complete set of MUBs: ◦ B = ew w F , Bm = bm,v v F , m F, ∞ { | ∈ } { | ∈ } ∈ 1 Tr(w ˆ (w ˆ mˆ )+2w ˆvˆ) bm,v = ω · ◦ ew. √q w F X∈ Proof. We show that for any m, v F , the vector ∈ Tr(w ˆ (w ˆ mˆ )+2w ˆvˆ) dm,v = ω · ◦ ew w F X∈ is an eigenvector of Du,u m for all u V . Indeed, ◦ ∈ Tr(w ˆ (w ˆ mˆ )+2w ˆvˆ+2w ˆ u[m) Du,u m(dm,v) = ω · ◦ · ◦ ew+u ◦ w F ∈ X \ \ Tr((w+u) ((w+u) mˆ )+2(w ˆ+ˆu)ˆv+2(w ˆ+ˆu) (ˆu mˆ )) = ω · ◦ · ◦ ew w F X∈ Tr((w ˆ+ˆu+2√wˆuˆ) ((w ˆ+ˆu+2√wˆuˆ) mˆ )+2(w ˆ+ˆu)ˆv+2(w ˆ+ˆu)(ˆu mˆ )) = ω · ◦ ◦ ew w F X∈ Tr(w ˆ (w ˆ mˆ )+2w ˆvˆ+S+ˆu(ˆu mˆ )+2ˆuvˆ+2ˆu(ˆu mˆ )) = ω · ◦ ◦ ◦ ew w F X∈ Tr(2ˆuvˆ uˆ(ˆu mˆ )) = ω − ◦ dm,v where Tr(S) = Tr((w ˆ (ˆu mˆ ) +u ˆ (w ˆ mˆ )+2ˆw (ˆu mˆ )) · ◦ · ◦ · ◦ +(2w ˆ (√wˆuˆ mˆ ) + 2√wˆuˆ (w ˆ mˆ )) · ◦ · ◦ +(2ˆu (√wˆuˆ mˆ ) + 2√wˆuˆ (ˆu mˆ ))) = 0 · ◦ · ◦ by Lemma 4.1.  Theorem 4.3 Let (F, +, ) be a commutative presemifield. Then the following forms a complete set of MUBs: ∗ B = ew w F , Bm = bm,v v F , m F, ∞ { | ∈ } { | ∈ } ∈ 1 Tr(m ˆ (w ˆ wˆ)+2w ˆvˆ) bm,v = ω ∗ ew. √q w F X∈ i i Proof. Let x y = a x2 y2 . Working as in section 3 we see that the corresponding ∗ ij symplectic presemifield (F, +, ) is given by multiplication: P ◦ r−i r−i r−i j−i r−i r−j r+i−j r−j x y = a2 xy2 + a2 x2 y2 + a2 x2 y2 . ◦ ii ij ij X Xi

13 Now, the statement of theorem follows from Theorem 4.2.  The above theorems allow us to construct complete sets of MUBs directly from presemifields (in particular, from Kantor-Williams [32] semifields).

4.1 Desarguesian spreads Desarguesian spreads are constructed with the help of finite fields, and the corresponding com- mutative and symplectic semifields are the same: x y = x y = xy. Theorem 2.2 and [37] ∗ ◦ imply that for the automorphism group of the corresponding complete set of MUBs we have

Aut( ) = F 2.(SL (q).Z ).Z , B ∼ 2 r 2 2 where the last factor Z2 is absent in the case q = 2. We note that the extension F .(SL2(q).Zr) is non-split for q 8 (see [1]), but the automorphism group of the corresponding plane is a ≥ semidirect product.

4.2 L¨uneburg planes and Suzuki groups In this subsection we consider non-semifield symplectic spreads related to L¨uneburg planes and Suzuki groups [34, 46]. We give an explicit construction of MUBs in terms of Galois rings. Let k+1 F = F be a finite field of order q = 22k+1 and let σ : α α2 be an automorphism of F , q → q V = F F . q ⊕ q The symplectic spread Σ of a space W = V V consists of the subspace (0,y) y V ⊕ { | ∈ } and subspaces (x,xM ) x V , c V , where { c | ∈ } ∈ −1 −1 α ασ + β1+σ c = (α, β) V, Mc = σ−1 1+σ−1 . ∈ α + β β ! The automorphism group of this symplectic spread is

Aut(Σ) = Aut(Sz(q)) = Sz(q).Z2k+1,

2 where Sz(q)= B2(q) is the Suzuki twisted simple group. C The corresponding orthogonal decomposition of the Lie algebra L = slq2 ( ) is given by

L = H c V Hc, ∞ ⊕ ∈

H = D0,y y V,y = 0 ,Hc = Dx,xMc x V,x = 0 . ∞ h | ∈ 6 i h | ∈ 6 i Let R = GR(42k+1) be a Galois ring of characteristic 4 and cardinality 42k+1. For v = α β (α, β) V we define the Teichm¨uller lift of v as (ˆα, βˆ), and for the matrix M = we ∈ γ δ   αˆ βˆ define the Teichm¨uller lift of M as Mˆ = . Then the following bases form a complete γˆ δˆ   set of MUBs: B = ew w V , Bc = bc,v v V , c V, ∞ { | ∈ } { | ∈ } ∈

1 Tr(w ˆ wˆMˆc+2ˆv wˆ) b = ω · · e . c,v q w w V X∈

14 Indeed,

c c 1 Tr(w ˆ wˆMc+2ˆv wˆ+2ˆuMc wˆ) D (b ) = ω · · · e u,uMc c,v q w+u w V X∈ \ \ c \ c \ 1 Tr((w+u) (w+u)Mc+2ˆv (w+u)+2ˆuMc (w+u)) = ω · · · e . q w w V X∈ Now we set w = (w1, w2), u = (u1, u2). Then \ (w + u) = (w1 + u1 + 2 w1u1, w2 + u2 + 2 w2u2) =w ˆ +u ˆ + 2ˆz, wherez ˆ = (√w u , √w u ). Therefore,p p 1 1 2 2c c cc c c cc c c 1 Tr((w ˆ+ˆu+2ˆz) (w ˆ+ˆu+2ˆz)Mc+2ˆv (w ˆ+ˆu)+2ˆuMc (w ˆ+ˆu) Dcu,uMcc (bc,vc)c = ω · · · ew q w V X∈ c c 1 Tr[(w ˆ wˆMc+2ˆv wˆ)+(3ˆu uˆMc+2ˆv uˆ)] = ω · · · · e q w w V X∈ c Tr(2ˆv uˆ uˆ uˆMc) = ω · − · bc,v, which shows that the vectors bc,v are common eigenvectors for the Cartan subalgebra Hc.

5 Pseudo-planar functions and MUBs

Let F = Fq be a finite field of even order q. There are no planar functions over fields of even characteristic. Recently Zhou [51, 42] introduced the notion of “planar” functions in even characteristic, however we adopt another term and call a function f : F F pseudo-planar if → x f(x + a)+ f(x)+ ax 7→ is a permutation of F for each a F . Using the Teichm¨uller lift, we can also consider f as a ∈ ∗ function f : . T →T

Theorem 5.1 Let F = Fq be a finite field of even characteristic and f be a pseudo-planar function. Then the following forms a complete set of MUBs:

B = ew w F , Bm = bm,v v F , m F, ∞ { | ∈ } { | ∈ } ∈ 1 Tr(m ˆ (w ˆ2+2f(w ˆ))+2ˆvwˆ) bm,v = ω ew, √q w F X∈ Proof. If m = m1 then

1 Tr(2(ˆv vˆ1)w ˆ) 1 if v = v′ (bm,v, bm1,v1 )= ω − = q 0 if v = v′. w F  6 X∈ If m = m then 6 1 2 1 Tr((m ˆ mc1)(w ˆ +2f(w ˆ))+2(ˆv vb1)w ˆ) (b , b ) = ω − − m,v m1,v1 q w F X∈ 1 2 = ωTr(M(w ˆ +2f(w ˆ))+2ˆuwˆ), q w F X∈ 15 where M =m ˆ mˆ 2R,u ˆ = v\v . Then − 1 6∈ − 1 2 2 2 1 Tr(M(w ˆ +2f(w ˆ))+2ˆuwˆ M((wc1) +2f(wc1)) 2ˆuwc1) (b , b ) = ω − − | m,v m1,v1 | q2 w,w1 F X∈ 2 2 1 Tr(M(w ˆ (wc1) +2f(w ˆ) 2f(wc1))+2ˆu(w ˆ wc1)) = ω − − − . q2 w,w1 F X∈ Let (w )2 wˆ2 +a ˆ2 (mod 2R), a F . Then (w )2 =w ˆ2 +a ˆ2 + 2w ˆaˆ. Therefore, 1 ≡ ∈ 1

2 1 Tr(M( aˆ2 2w ˆaˆ+2f(w ˆ) 2f(w[+a)) 2ˆuaˆ) c (b , b ) = ω c− − − − | m,v m1,v1 | q2 w,a F X∈ 1 Tr( Maˆ2 2ˆuaˆ) Tr(2M(f(w[+a)+f(w ˆ)+w ˆaˆ)) = ω − − ω . q2 a F w F X∈ X∈ Since f(w) is a pseudo-planar function, we have

[ q if a = 0 ωTr(2M(f(w+a)+f(w ˆ)+w ˆaˆ)) = 0 if a = 0. w F  6 X∈ Hence, 1 1 (b , b ) 2 = q = .  | m,v m1,v1 | q2 · q The following theorem shows connections between pseudo-planar functions and commutative presemifields.

Theorem 5.2 Let F be a finite field of characteristic two. 1. If (F, +, ) is a commutative presemifield with multiplication given by ∗ i j j i x y = xy + a (x2 y2 + x2 y2 ) ∗ ij Xi

Proof. 1. If x y = xy + f(x + y) + f(x) + f(y) determines a presemifield then the ∗ map x xy + f(x + y)+ f(x)+ f(y) is a permutation for any y F , therefore the map 7→ ∈ ∗ x xy + f(x + y)+ f(x) is a permutation for any y F . 7→ ∈ ∗ 2. Assume that the presemifield (F, +, ) is given by ∗ i i i j j i x y = a x2 y2 + a (x2 y2 + x2 y2 ). ∗ i ij X Xi

16 Suppose that g is not invertible. Then there exists a F such that g(a2) = 0. Then ∈ ∗ (x + a) (x + a)= g((x + a)2)= g(x2 + a2)= g(x2)= x x, ∗ ∗ which means x x + x a + a x + a a = x x and a a = 0, a contradiction. ∗ ∗ ∗ ∗ ∗ ∗ 2. The condition (x + z) y = x y + z y is equivalent to ∗ ∗ ∗ (x + z)y + f(x + z + y)+ f(x + z)+ f(y)

= xy + f(x + y)+ f(x)+ f(y)+ zy + f(z + y)+ f(z)+ f(y). Therefore,

f(x + y + z)+ f(x + y)+ f(x + z)+ f(y + z)+ f(x)+ f(y)+ f(z) = 0, which means that f is a quadratic function.  Remark 2. The notion of pseudo-planar functions can be defined over a field F of arbitrary characteristic. We call a function f : F F pseudo-planar if the map x f(x + a) f(x)+ ax → 7→ − is a permutation of F for each a F . Such functions carry similar properties as pseudo- ∈ ∗ planar functions in even characteristic (including MUBs constructions). In particular, if f is a quadratic pseudo-planar function then the product x y = xy + f(x + y) f(x) f(y) defines ∗ − − a commutative presemifield (F, +, ). Probably such a definition of pseudo-planar functions ∗ provides unified approach to all characteristics, at least it provides a bridge between the even and odd characteristic cases. Note that in the case of odd characteristic function f is pseudo- planar if and only if the function x2 + 2f(x) is planar, and formulas in Theorems 3.4 and 5.1 have a unified look in the language of pseudo-planar functions.

Acknowledgments The author would like to thank William M. Kantor, Claude Carlet and Yue Zhou for valuable discussions on the content of this paper. This research was supported by UAEU grant 21S073 and NRF grant 31S088.

References

[1] K. S. Abdukhalikov, On a group of automorphisms of an orthogonal decomposition of a Lie algebra of type A2m 1, (Russian) Izv. Vyssh. Uchebn. Zaved. Mat., no. 10, 11–14 (1991); − English translation in Soviet Math. (Iz. VUZ) 35, 9–12 (1991).

[2] K. S. Abdukhalikov, Invariant integral lattices in Lie algebras of type Apm 1, (Russian) − Mat. Sb. 184, no. 4, 61–104 (1993); English translation in Russian Acad. Sci. Sb. Math. 78, no. 2, 447–478 (1994).

[3] K. S. Abdukhalikov, Affine invariant and cyclic codes over p-adic numbers and finite rings, Des. Codes Cryptogr. 23, 343–370 (2001).

[4] K. Abdukhalikov, E. Bannai and S. Suda, Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets, J. Combin. Theory Ser. A 116, 434–448 (2009).

[5] W. O. Alltop, Complex sequences with low periodic correlations, IEEE Trans. Inform. Theory 26 (3), 350–354 (1980).

17 [6] L. Bader, W. M. Kantor and G. Lunardon, Symplectic spreads from twisted fields, Boll. Un. Math. Ital. A (7) 8, 383–389 (1994).

[7] S. Ball, J. Bamberg, M. Lavrauw and T. Penttila, Symplectic spreads, Des. Codes Cryptogr. 32, 9–14 (2004).

[8] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, A new proof for the existence of mutually unbiased bases, Algorithmica 34, 512–528 (2002).

[9] C. H. Benneth and G. Brassard, Quantum Cryptography: Public key distribution and coin tossing, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore 175 (1984).

[10] J. Bierbrauer, New semifields, PN and APN functions, Des. Codes Cryptogr. 54(3), 189–200 (2010).

[11] P. O. Boykin, M. Sitharam, Pham Huu Tiep and P. Wocjan, Mutually unbiased bases and orthogonal decompositions of Lie algebras, Quantum Inf. Comput. 7, no. 4, 371–382 (2007).

[12] L. Budaghyan and T. Helleseth, New perfect nonlinear multinomials over Fp2k for any odd prime p. In: SETA 08: Proceedings of the 5th International Conference on Sequences and their Applications, 403–414. Springer, Berlin, Heidelberg (2008).

[13] A. R. Calderbank, P. J. Cameron, W. M. Kantor and J. J. Seidel, Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets, Proc. London Math. Soc. (3) 75, 436–480 (1997).

[14] S. D. Cohen, M. J. Ganley, Commutative semifields, two-dimensional over their middle nuclei, J. Algebra 75, 373–385 (1982).

[15] R. S. Coulter and R. W. Matthews, Planar functions and planes of Lenz-Barlotti class II, Des. Codes Cryptogr. 10, 167–184 (1997).

[16] R. S. Coulter, M. Henderson and P. Kosick, Planar polynomials for commutative semifields with specified nuclei, Des. Codes Cryptogr. 44, 275–286 (2007).

[17] R. S. Coulter and P. Kosick, Commutative semifields of order 243 and 3125, in: Finite Fields: Theory and Applications, in: Contemp. Math., vol. 518, Amer. Math. Soc., Providence, RI, 129–136 (2010).

[18] P. Dembowski, Finite geometries, Springer, Berlin (1968).

[19] L. E. Dickson, On commutative linear algebras in which division is always uniquely possible, Trans. Amer. Math. Soc. 7, 514–522 (1906).

[20] C. Ding and J. Yuan, A family of skew Hadamard difference sets, J. Combin. Theory Ser. A 113, 1526–1535 (2006).

[21] C. Ding and J. Yin, Signal sets from functions with optimum nonlinearity, IEEE Trans. Communications 55, No. 5, 936–940 (2007).

[22] M. J. Ganley, Central weak nucleus semifields, European J. Combin. 2, 339–347 (1981).

[23] C. Godsil and A. Roy, Equiangular lines, mutually unbiased bases, and spin models, Euro- pean J. Combin. 30, 246–262 (2009).

18 [24] R. Gow, Generation of mutually unbiased bases as powers of a unitary matrix in 2-power dimensions, arXiv:math/0703333.

[25] R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Sol´e, The Z4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40, No. 2, 301–319 (1994).

[26] I. D. Ivanovi´c, Geometrical description of quantal state determination, J. Phys. A 14, 3241– 3245 (1981).

[27] W. M. Kantor and M. E. Williams, New flag-transitive affine planes of even order, J. Comb. Theory(A) 74, 1–13 (1996).

[28] W. M. Kantor, Ovoids and translation planes, Canad. J. Math. 34, 1195–1207 (1982).

[29] W. M. Kantor, Projective planes of order q whose collineation groups have order q2, J. Alg. Comb. 3, 405–425 (1994).

[30] W. M. Kantor, Note on Lie algebras, finite groups and finite geometries, in: Groups, Difference Sets, and the Monster (Eds. K. T. Arasu et al.) 73–81, de Gruyter, Berlin-New York (1996).

[31] W. M. Kantor, Commutative semifields and symplectic spreads, J. Algebra 270, 96–114 (2003).

[32] W. M. Kantor and M. E. Williams, Symplectic semifield planes and Z4-linear codes, Trans. Amer. Math. Soc. 356, 895–938 (2004).

[33] W. M. Kantor and M. E. Williams, Nearly flag-transitive affine planes, Adv. Geom. 10, 161–183 (2010).

[34] W. M. Kantor, MUBs inequivalence and affine planes, J. Math. Phys. 53, 032204 (2012).

[35] A. Klappenecker and M. R¨otteler, Constructions of mutually unbiased bases, in: Finite Fields and Applications, Lecture Notes in Computer Sciences, vol. 2948, 137–144, Springer, Berlin (2004).

[36] D. E. Knuth, Finite semifields and projective planes, J. Algebra 2, 182–217 (1965).

[37] A. I. Kostrikin and Pham Huu Tiep, Orthogonal decompositions and integral lattices, Walter de Gruyter, Berlin (1994).

[38] N. LeCompte, W. J. Martin and W. Owens, On the equivalence between real mutually unbiased bases and a certain class of association schemes, European J. Combin., 31, 1499– 1512 (2010).

[39] A. A. Nechaev, Kerdock code in a cyclic form, (Russian) Diskret. Mat. 1, no. 4, 123–139 (1989); English translation in Discrete Math. Appl. 1, no. 4, 365–384 (1991).

[40] T. Penttila and B. Williams, Ovoids of parabolic spaces. Geom. Dedicata 82, 1–19 (2000).

[41] A. Roy and A. J. Scott, Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements, J. Math. Phys. 48, 072110 (2007).

[42] K.-U. Schmidt and Y. Zhou, Planar functions over fields of characteristic two, J. Algebraic Combin. 40, 503–526 (2014).

19 [43] J. Schwinger, Unitary operator bases, Proc. Nat. Acad. Sci. U.S.A. 46, 570-579 (1960).

[44] C. Suetake, A new class of translation planes of order q3, Osaka J. Math. 22, 773–786 (1985).

[45] J. A. Thas and S. E. Payne, Spreads and ovoids in finite generalized quadrangles, Geom. Dedicata 52, 227–253 (1994).

[46] J. Tits, Ovo¨ıdes et groupes de Suzuki, Arch. Math. 13, 187–198 (1962).

[47] G. Weng and X. Zeng, Further results on planar DO functions and commutative semifields, Des. Codes Cryptogr. 63, 413–423 (2012).

[48] W. K. Wootters and B. D. Fields, Optimal state-determination by mutually unbiased mea- surements, Ann. Physics 191 (2), 363–381 (1989).

[49] Z. Zha, G. M. Kyureghyan and X. Wang, Perfect nonlinear binomials and their semifields, Finite Fields and Their Applications 15, 125–133 (2009).

[50] Z. Zha and X. Wang, New families of perfect nonlinear polynomial functions, J. Algebra 322, 3912–3918 (2009).

[51] Y. Zhou, (2n, 2n, 2n, 1)-relative difference sets and their representations, J. Combin. Designs 21 (12), 563–584 (2013).

[52] Y. Zhou and A. Pott, A new family of semifields with 2 parameters, Adv. Math. 234, 43–60 (2013).

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