Finite Geometry: Pure Mathematics Close to Practical Applications

Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry

Finite geometry: Pure mathematics close to practical applications

Leo Storme

Ghent University Dept. of Mathematics: Analysis, Logic and Discrete Mathematics Krijgslaan 281 - Building S8 9000 Ghent Belgium

Francqui Foundation, April 19, 2021

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry OUTLINE

1 FINITE GEOMETRY

2 ARCS IN FINITE PROJECTIVE SPACES

3 FURTHER APPLICATIONS OF FINITE GEOMETRY

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry OUTLINE

1 FINITE GEOMETRY

2 ARCS IN FINITE PROJECTIVE SPACES

3 FURTHER APPLICATIONS OF FINITE GEOMETRY

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FINITE GEOMETRY

What is finite geometry? Work with finite structures. Investigation of geometrical properties. Examples: finite projective spaces, finite generalized quadrangles, partial geometries, semi-partial geometries, ..., Some problems of other research areas can be translated into problems regarding substructures in finite geometries.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FINITE FIELDS

q = prime number.

Prime ﬁelds: Fq = {0,..., q − 1} (mod q). Calculating (mod q) means calculating with the integers 0,..., q − 1, but add/subtract multiples of q till the result again is an integer in {0,..., q − 1}.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FINITE FIELDS

Example: q = 7. F7 = {0,..., 6} (mod 7). 2 + 3 ≡ 5 (mod 7). 5 + 3 ≡ 1 (mod 7). 5 · 3 ≡ 1 (mod 7). Calculating with the days of the week is calculating (mod 7). If today is Thursday, which day of the week are we in ﬁve days?

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FINITE FIELDS

Finite ﬁelds (Galois ﬁelds): Fq: exist for every prime power q, q = ph, p prime, h ≥ 1.

Évariste Galois

(25 oktober 1811, Bourg-la-Reine - 31 mei 1832, Parijs)

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry 3-DIMENSIONAL VECTOR SPACE

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry VECTOR SPACES OVER FINITE FIELDS

Vector space V (n, q) of dimension n over the ﬁnite ﬁeld Fq of order q

V (n, q) = {(x1,..., xn): x1,..., xn ∈ Fq} (x1,..., xn) + (y1,..., yn) = (x1 + y1,..., xn + yn). α(x1,..., xn) = (αx1, . . . , αxn), for α ∈ Fq.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry ADDITION AND SCALAR MULTIPLICATION OF VECTORS

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FROM VECTOR SPACE TO PROJECTIVE SPACE

For some problems regarding vector spaces, it is allowed to replace a non-zero vector (x1,..., xn) by a non-zero scalar multiple α(x1,..., xn), α ∈ Fq \{0}. In ﬁnite geometry, a non-zero vector (x1,..., xn) and its non-zero scalar multiple α(x1,..., xn), α ∈ Fq \{0}, are considered as one object, called a projective point.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FROM VECTOR SPACE TO PROJECTIVE SPACE

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry THE FANOPLANE PG(2, 2)

From V (3, 2) to PG(2, 2)

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry THE FANOPLANE PG(2, 2)

Gino Fano (1871-1952)

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry PROJECTIVEPLANEOFPRIMEORDER

Let p be a prime, the projective plane PG(2, p) contains p2 + p + 1 points p2 points (1, y, z), y, z ∈ {0, 1,..., p − 1} (mod p), p points (0, 1, z), z ∈ {0, 1,..., p − 1} (mod p), one point (0, 0, 1). p2 + p + 1 lines p2 lines X + aY + bZ = 0, a, b ∈ {0, 1,..., p − 1} (mod p), p lines Y + bZ = 0, b ∈ {0, 1,..., p − 1} (mod p), one line Z = 0.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry POINTBELONGSTOALINE

Point of PG(2, p) belongs to line of PG(2, p) if and only if coordinates of the point satisfy the linear equation deﬁning the line. Examples (1, y, z) belongs to the line X + aY + bZ = 0 if and only if 1 + ay + bz = 0 (mod p). (0, 0, 1) belongs to the line X + aY + bZ = 0 if and only if 0 + a · 0 + b · 1 = b = 0 (mod p).

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry THE FANOPLANE PG(2, 2)

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry PROPERTIESOFTHEPROJECTIVEPLANE PG(2, p)

Properties of the projective plane PG(2, p), p prime: a point belongs to p + 1 lines, and two points belong to one common line. a line contains p + 1 points, and two lines intersect in one common point. there exist four points in PG(2, p), no three on a common line.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry THEPLANE PG(2, 2)

The plane PG(2, 2) contains 7 points and 7 lines. A point belongs to 3 lines and a line contains 3 points.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry THEPLANE PG(2, 3)

The plane PG(2, 3) contains 13 points and 13 lines. A point belongs to 4 lines and a line contains 4 points.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry THEPLANE PG(2, 5)

The plane PG(2, 5) contains 31 points and 31 lines. A point belongs to 6 lines and a line contains 6 points.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry THEPLANE PG(2, 7)

The plane PG(2, 7) contains 57 points and 57 lines. A point belongs to 8 lines and a line contains 8 points.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry PROPERTIESOFTHEPROJECTIVEPLANE PG(2, p)

Every two lines intersect in one common point. No parallel lines in PG(2, p), p prime. Idea: parallel lines share a point at inﬁnity.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry PARALLEL LINES SHARE A POINT AT INFINITY

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry THEPLANE PG(2, 3)

The plane PG(2, 3) contains 13 points and 13 lines. A point belongs to 4 lines and a line contains 4 points. Points 10, 11, 12 and 13 are points at inﬁnity.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FROM V (n + 1, q) TO PG(n, q)

1 From V (1, q) to PG(0, q) (projective point), 2 From V (2, q) to PG(1, q) (projective line), 3 ··· 4 From V (i + 1, q) to PG(i, q) (i-dimensional projective subspace), 5 ··· 6 From V (n, q) to PG(n − 1, q) ((n − 1)-dimensional subspace = hyperplane), 7 From V (n + 1, q) to PG(n, q) (n-dimensional space).

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry PG(3, 2)

From V (4, 2) to PG(3, 2)

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry OUTLINE

1 FINITE GEOMETRY

2 ARCS IN FINITE PROJECTIVE SPACES

3 FURTHER APPLICATIONS OF FINITE GEOMETRY

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry BENIAMINO SEGRE

(February 16, 1903, Turin - October 2, 1977, Frascati)

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry BENIAMINO SEGRE

Beniamino Segre: one of the founding fathers of ﬁnite geometry. Stimulated research on ﬁnite geometry: Finite geometry is an interesting research area.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry DEFINITION

DEFINITION (SEGRE) n-Arc in PG(k − 1, q): set of n points, every k linearly independent.

Example: n-arc in PG(2, q): n points, no three collinear.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry NORMAL RATIONAL CURVE

Normal rational curve (NRC) = classical example of (q + 1)-arc in PG(k − 1, q):

k−1 {(1, t,..., t )||t ∈ Fq} ∪ {(0,..., 0, 1)}. For k − 1 = 2, NRC = conic

2 {(1, t, t )||t ∈ Fq} ∪ {(0, 0, 1)} 2 is set of points of conic X1 = X0X2.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry VANDERMONDEMATRIXANDDETERMINANT

1 1 ··· 1

t1 t2 ··· tk

t2 t2 ··· t2 Y 1 2 k = (tj − ti ) 6= 0, ...... j>i k−1 k−1 k−1 t1 t2 ··· tk

for k distinct elements t1,..., tk ∈ Fq. (Alexandre Théophile Vandermonde (1735-1796))

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FUNDAMENTAL THEOREM OF FINITE GEOMETRY

For k − 1 = 2, NRC = conic

2 {(1, t, t )||t ∈ Fq} ∪ {(0, 0, 1)} 2 is set of points of conic X1 = X0X2. Conic in PG(2, q) is classical example of a (q + 1)-arc. B. Segre considered the following question: Is every (q + 1)-arc in PG(2, q) a conic?

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FUNDAMENTAL THEOREM OF FINITE GEOMETRY

THEOREM (FUNDAMENTAL THEOREM OF FINITE GEOMETRY (SEGRE)) Every (q + 1)-arc in PG(2, q), q odd, is a conic.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FURTHER RESULTS

THEOREM (JOSEPH A.THAS) Every (q + 1)-arc in PG(k − 1, q), q odd prime power, k small, is a normal rational curve.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FURTHER RESULTS

THEOREM (SIMEON BALL) There are no (q + 2)-arcs in PG(k − 1, q), q odd prime.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry CODING THEORY

During transmission of information between computers, from a space probe to the earth, errors occur. Some symbols are received incorrectly. Scratches on CD and DVD make bits unreadable. Coding theory designs codes to correct errors after transmission. Basic principle of coding theory: transmit more bits than strictly necessary.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry CODING THEORY

Example: Transmit YES or NO via computer. Solution 1: YES = 0 and NO = 1. Disadvantage to solution 1: no errors are allowed during transmission. Solution 2: YES = 00000 and NO = 11111. Advantage of solution 2: up to two bits can arrive incorrectly, correct transmitted codeword can be found.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry CODING THEORY

Coding theory Research area that construct Codes for the transmission of information between computers, from space probes to the earth, storage of music on CD, movie on DVD, storage of information on QR-codes (Quick Response codes).

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry LINEARCODES

Linear codes are deﬁned by k × n matrix that converts a sequence of k symbols into a sequence of n symbols. Example: Binary matrix

 1 0 1 0 1 0 1  G =  0 1 1 0 0 1 1  0 0 0 1 1 1 1

converts binary sequences of length 3 (messages) into binary sequences of length 7 (codewords).

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry LINEARCODES

Example: Binary matrix

 1 0 1 0 1 0 1  G =  0 1 1 0 0 1 1  0 0 0 1 1 1 1

converts binary sequences of length 3 (message) into binary sequences of length 7 (codewords).

Message m = (m1, m2, m3) becomes codeword m·G = (m1, m2, m1+m2, m3, m1+m3, m2+m3, m1+m2, m3).

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry MAXIMUM DISTANCE SEPARABLE CODES

The codes with the best error-correcting capacities, are the Maximum Distance Separable (MDS) codes. Classical example: Reed-Solomon codes (RS-codes), deﬁned by matrix .  1 1 ··· 1 0   t1 t2 ··· tq 0   2 2 2   t t ··· tq 0   1 2  G =  . . . .  ,  . . ··· . .   k−2 k−2 k−2   t1 t2 ··· tq 0  k−1 k−1 k−1 t1 t2 ··· tq 1

with Fq = {t1,..., tq}.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry REED-SOLOMONCODES

(Irving S. Reed and Gustave Solomon)

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry CODINGTHEORETICALQUESTION

Question: are there other linear codes, that can correct the same number of errors as the Reed-Solomon codes? Search for other k × (q + 1) matrices   g11 g12 ··· g1q g1,q+1  g21 g22 ··· g2q g2,q+1     g31 g32 ··· g3q g3,q+1  G =    . . . .   . . ··· . .     gk−1,1 gk−1,2 ··· gk−1,q gk−1,q+1  gk,1 gk,2 ··· gk,q gk,q+1

with the property that every k columns in G are linearly independent.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry GEOMETRICALLY EQUIVALENT QUESTION

Question: are there other linear codes, that can correct the same number of errors as the Reed-Solomon codes? Is equivalent to: ﬁnd an other matrix G with the property that every k columns in G are linearly independent. Geometrical question: Can you ﬁnd q + 1 points in the projective space PG(k − 1, q), every k linearly independent, different from a normal rational curve?

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry DEFINITION

DEFINITION (SEGRE) n-Arc in PG(k − 1, q): set of n points, every k linearly independent.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry ARCS

Equivalence:

MDS codes equivalent with Arcs in ﬁnite projective spaces (Segre)

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry LINKBETWEENLINEAR MDS CODESANDARCS

THEOREM Linear [n, k, d]-code C is linear MDS code if and only if the n columns of a generator matrix G of C form an n-arc in PG(k − 1, q).

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry BENIAMINO SEGRE

Beniamino Segre (1903 - 1977)

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FUNDAMENTAL THEOREM OF FINITE GEOMETRY

Beniamino Segre proved

THEOREM Every (q + 1)-arc in PG(2, q), q odd, is a normal rational curve.

This theorem implied the following theorem from coding theory.

THEOREM Every Maximum Distance Separable code of length q + 1 and dimension 3 over a ﬁnite ﬁeld of order q, q odd, is a Reed-Solomon code.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FURTHER RESULTS

THEOREM (JOSEPH A.THAS) Every (q + 1)-arc in PG(k − 1, q), q odd, k small, is a normal rational curve. This theorem implied the following theorem from coding theory.

THEOREM Every Maximum Distance Separable code of length q + 1 and dimension k, k small, over a ﬁnite ﬁeld of order q, q odd, is a Reed-Solomon code.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry FURTHER RESULTS

THEOREM (SIMEON BALL) There are no (q + 2)-arcs in PG(k − 1, q), q odd prime.

This theorem implied the following theorem from coding theory.

THEOREM There do not exist Maximum Distance Separable codes of length q + 2 and dimension k over a ﬁnite ﬁeld of order q, q odd prime.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry APPLICATIONS OF REED-SOLOMONCODES

music CD uses a [28, 24, 5] RS-code and a [32, 28, 5] RS-code over the finite field F256 = F28 . CCSDS standard (Consultive Committee for Space Data Systems) uses a [255, 223, 33] RS-code over the finite field F256 = F28 . DVD and QR-codes (Quick Response) use RS-codes.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry CCSDS STANDARD

CCSDS standard (Consultive Committee for Space Data Systems)

Fig. 6. CCSDS Standard for deep space telemetry links.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry ENCODINGOFMUSIC

6 message = 44100 -second = 24 bytes = mt = (mt,1,..., mt,24) message mt encoded in codeword ct = (ct,1,..., ct,28) of [28, 24, 5] RS-code over F256, 4-frame delayed interleaving

c1,1 ··· c5,1 ··· c9,1 ··· c109,1 ··· ······ c1,2 ··· c5,2 ··· c105,2 ··· ············ c1,3 ··· c101,3 ··· ...... ·················· c1,28 ···

column = message of [32, 28, 5] RS-code over F256, column t encoded in codeword dt = (dt,1,..., dt,32). scratches of approximately 4000 bits = 2.5mm can be corrected. Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry QR-CODE

QR-code = Quick Response code

Position Detecrion \ \ Pattems

Separators loc Position Fuuctioti Detecrion Pattems Pattems Sjmbol Timing Parten»

Aliamnent Pattems /

\ Fonnat Intonnation

Version Infonnarion Eneedins Reaioti

Data and Eiror Correction Codewords / /

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry QR-CODE

QR-code = Quick Response code □

□

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry MANY FURTHER LINKS BETWEEN FINITE GEOMETRY ANDOTHERRESEARCHDOMAINS

Coding theory Arcs and MDS codes, Minihypers and linear codes meeting the Griesmer bound, Subspace codes, Cooling codes. Cryptography. Graph theory.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry OUTLINE

1 FINITE GEOMETRY

2 ARCS IN FINITE PROJECTIVE SPACES

3 FURTHER APPLICATIONS OF FINITE GEOMETRY

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry DOBBLE

Card game Dobble

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry DOBBLE

Card game Dobble

Based on the projective plane PG(2, 7)

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry THEPLANE PG(2, 7)

The plane PG(2, 7) contains 57 points and 57 lines. A point belongs to 8 lines and a line contains 8 points.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry PLAY DOBBLE

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry SET

Card game SET

Based on the 4-dimensional affine space AG(4, 3) over the finite field of order 3.

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry THE FANOTUNE

The Fano tune by Sam Adriaensen. https://soundcloud.com/sam_a3/the-fano-tune May not music be described as the mathematics of the sense, mathematics as music of the reason? The musician feels mathematics, the mathematician thinks music: music the dream, mathematics the working life. - James Joseph Sylvester

Leo Storme Finite geometry Finite geometry Arcs in ﬁnite projective spaces Further applications of ﬁnite geometry

Thank you very much for your attention!

Leo Storme Finite geometry