Linear codes and substructures of finite geometry

From coding theory to Galois geometriesLinear MDS codes and arcs

Griesmer bound and minihypersCovering radius and saturating sets

Linear codes from finite projective spaces

Linear codes and substructures of finite geometry

Leo Storme

Ghent UniversityDept. of Mathematics: Analysis, Logic and Discrete Mathematics

Krijgslaan 281 - Building S89000 Ghent

Belgium

Francqui Foundation, April 22, 2021

Leo Storme Linear codes and substructures of finite geometry




OUTLINE

1 FROM CODING THEORY TO GALOIS GEOMETRIES

2 LINEAR MDS CODES AND ARCS

3 GRIESMER BOUND AND MINIHYPERS

4 COVERING RADIUS AND SATURATING SETS

5 LINEAR CODES FROM FINITE PROJECTIVE SPACESThe case PG(2,p), p primeDual code of PG(2,q)





LINEAR CODES

q = prime number,Prime fields: Fq = {1, . . . ,q} (mod q),Finite fields (Galois fields): Fq: q prime power,Linear [n, k ,d ]-code C over Fq is:

k -dimensional subspace of V (n,q),minimum (Hamming) distance d = minimal number ofpositions in which two distinct codewords differ.

If d = 2t + 1 or d = 2t + 2, then C is t-error correcting.





LINEAR CODES

Generator matrix of [n, k ,d ]-code C:

G = (g1 · · · gn)

G = (k × n) matrix of rank k ,rows of G form basis of C,codeword of C = linear combination of rows of G.





LINEAR CODES

Parity check matrix H for C:(n − k)× n matrix of rank n − k ,c ∈ C ⇔ c · HT = 0.





REMARK

Remark: For linear [n, k ,d ]-code C, fundamental parametersn, k ,d do not change when column gi in generator matrix

G = (g1 · · · gn)

is replaced by non-zero scalar multiple.





FROM VECTOR SPACE TO PROJECTIVE SPACE





THE FANO PLANE PG(2,2)

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





THE FANO PLANE PG(2,2)

Gino Fano (1871-1952)





THE PLANE PG(2,3)





PG(3,2)

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





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).





OUTLINE










LINEAR MDS CODES AND ARCS

Question:Given

length n,dimension k ,

find maximal value of d .Result: Singleton (upper) bound

d ≤ n − k + 1.

Notation: MDS code = [n, k ,n − k + 1]-code.





ARCS

Equivalence:

Singleton (upper) bound (MDS codes)equivalent with

Arcs in finite projective spaces (Segre)





BENIAMINO SEGRE

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





DEFINITION

DEFINITION

n-Arc in PG(k − 1,q): set of n points, every k linearlyindependent.

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





NORMAL RATIONAL CURVE

Normal rational curve (NRC) = classical example of(q + 1)-arc:

{(1, t , . . . , tk−1)||t ∈ Fq} ∪ {(0, . . . ,0,1)}

For k − 1 = 2, NRC = conic

{(1, t , t2)||t ∈ Fq} ∪ {(0,0,1)}

is set of points of conic X 21 = X0X2.





FUNDAMENTAL THEOREM OF FINITE GEOMETRY

For k − 1 = 2, NRC = conic

{(1, t , t2)||t ∈ Fq} ∪ {(0,0,1)}

is set of points of conic X 21 = X0X2.

B. Segre considered the following question:Is every (q + 1)-arc in PG(2,q) a conic?






THEOREM (FUNDAMENTAL THEOREM OF FINITE GEOMETRY

(SEGRE))

Every (q + 1)-arc in PG(2,q), q odd, is a conic.

THEOREM (SEGRE)

Every n-arc in PG(2,q), q odd, with n > q −√q/4 + 25/16, is asubset of a conic.





CHARACTERIZATION RESULT

THEOREM (SEGRE, THAS)

In PG(k − 1,q), q odd, every n-arc, withn > q −√q/4 + k − 23/16, is a subset of a normal rationalcurve.

COROLLARY

In PG(k − 1,q), q odd, 3 ≤ k <√

q/4 + 39/16, every(q + 1)-arc is a normal rational curve.





HYPEROVALS IN PG(2,q), q EVEN

Situation in PG(2,q), q even, is completely different.Point (0,1,0) extends {(1, t , t2)||t ∈ Fq} ∪ {(0,0,1)} to a(q + 2)-arc.(q + 2)-arc in PG(2,q), q even, is called a hyperoval.Different classes of hyperovals in PG(2,q), q even, known.Classification of hyperovals in PG(2,q), q even: veryimportant, but very hard, problem.





REED-SOLOMON CODES

Linear MDS code = [n, k ,n − k + 1]-code.Classical example of linear MDS code =[q + 1, k ,d = q + 2− k ]-GDRS (GeneralizedDoubly-Extended Reed-Solomon) code with generatormatrix

G =

1 · · · 1 0t1 · · · tq 0...

......

...tk−21 · · · tk−2

q 0tk−11 · · · tk−1

q 1

Columns of G are points of a normal rational curve.





LINK BETWEEN LINEAR MDS CODES AND ARCS

THEOREM

Linear [n, k ,d ]-code C is linear MDS code if and only if ncolumns of a generator matrix G of C form n-arc inPG(k − 1,q).






THEOREM (FUNDAMENTAL THEOREM OF FINITE GEOMETRY

(SEGRE))

Every (q + 1)-arc in PG(2,q), q odd, is a conic.

THEOREM

Every linear [q + 1,3,q−1]-MDS code, q odd, is a GDRS code.






COROLLARY

In PG(k − 1,q), q, odd, 3 ≤ k <√

q/4 + 39/16, every(q + 1)-arc is a normal rational curve.

THEOREM (SEGRE, THAS)For

q odd prime power,3 ≤ k <

√q/4 + 39/16,

[n = q + 1, k ,d = q + 2− k ]-MDS code is GDRS.





BALL RESULT

THEOREM (BALL)

For q odd prime, n ≤ q + 1 for every [n, k ,n− k + 1]-MDS code.

Technique: Polynomial techniques





NON-EXTENDABILITY OF GDRS CODES

THEOREM (STORME AND THAS)

[q + 1, k ,q − k + 2]-GDRS code with generator matrix

G =

1 · · · 1 0t1 · · · tq 0...

......

...tk−21 · · · tk−2

q 0tk−11 · · · tk−1

q 1

.

For q odd prime power and 2 ≤ k ≤ q − 6√

q log q,[q + 1, k ,q + 2− k ]-GDRS code not extendable to[q + 2, k ,q + 3− k ]-MDS code.





APPLICATIONS OF REED-SOLOMON CODES

music CD uses a [28,24,5]-GDRS code and a[32,28,5]-GDRS code over the finite field of order256 = 28.CCSDS standard (Consultive Committee for Space DataSystems) uses a [255,223,33]-GDRS code over the finitefield of order 256 = 28.DVD and QR-codes (Quick Response) use GDRS-codes.





CCSDS STANDARD

CCSDS standard (Consultive Committee for Space DataSystems)

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





ENCODING OF MUSIC

message = 644100 -second = 24 bytes = mt = (m24t , . . . ,m24t+23)

message mt encoded in codeword ct = (ct ,1, . . . , ct ,28) of[28,24,5] GRS 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] GRS code over F256,column t encoded in codeword dt = (dt ,1, . . . ,dt ,32).

scratches of approximately 4000 bits = 2.5mm can becorrected.





OUTLINE










GRIESMER BOUND AND MINIHYPERS

Question: Givendimension k ,minimum distance d ,

find minimal length n of [n, k ,d ]-code over Fq.Result: Griesmer (lower) bound

n ≥k−1∑i=0

⌈dqi

⌉= gq(k ,d).





MINIHYPERS AND GRIESMER BOUND

Equivalence: (Hamada and Helleseth)

Griesmer (lower) boundequivalent with

minihypers in finite projective spaces





DEFINITION

DEFINITION

{f ,m; k − 1,q}-minihyper F is:set of f points in PG(k − 1,q),F intersects every (k − 2)-dimensional space in at least mpoints.

(m-fold blocking sets with respect to the hyperplanes ofPG(k − 1,q))






Let C = [gq(k ,d), k ,d ]-code over Fq.If k × n generator matrix

G = (g1 · · · gn),

minihyper = PG(k − 1,q) \ {g1, . . . ,gn}.





EXAMPLE

Example: Griesmer [8,4,4]-code over F2

G =

1 1 1 1 1 1 1 10 0 0 0 1 1 1 10 0 1 1 0 0 1 11 0 1 0 1 0 1 0

minihyper = PG(3,2)\ {columns of G} = plane (X0 = 0).





CORRESPONDING MINIHYPER





OTHER EXAMPLES

Example 1. Subspace PG(µ,q) in PG(k − 1,q) = minihyper of[n = (qk − qµ+1)/(q − 1), k ,qk−1 − qµ]-code (McDonald code).





BOSE-BURTON THEOREM

THEOREM (BOSE-BURTON)

A minihyper consisting of |PG(µ,q)| points intersecting everyhyperplane in at least |PG(µ− 1,q)| points is equal to aµ-dimensional space PG(µ,q).





RAJ CHANDRA BOSE

R.C. Bose and R.C. Burton, A characterization of flat spaces ina finite geometry and the uniqueness of the Hamming and theMcDonald codes. J. Combin. Theory, 1:96-104, 1966.





OTHER EXAMPLES

Example 2. t < q pairwise disjoint subspaces PG(µ,q)i ,i = 1, . . . , t , in PG(k − 1,q) = minihyper of[n = (qk − 1)/(q− 1)− t(qµ+1− 1)/(q− 1), k ,qk−1− tqµ]-code.






THEOREM (GOVAERTS AND STORME)

For q odd prime and 1 ≤ t ≤ (q + 1)/2,[n = (qk − 1)/(q − 1)− t(qµ+1 − 1)/(q − 1), k ,qk−1 − tqµ]-codeC: minihyper is union of t pairwise disjoint PG(µ,q).

Proof: uses results on blocking sets in PG(k − 1,q), q oddprime.





OUTLINE










DEFINITION

DEFINITION

Let C be linear [n, k ,d ]-code over Fq. The covering radius of Cis smallest integer R such that every n-tuple in Fn

q lies atHamming distance at most R from codeword in C.

THEOREM

Let C be linear [n, k ,d ]-code over Fq with parity check matrix

H = (h1 · · · hn).

Then covering radius of C is equal to R if and only if every(n − k)-tuple over Fq can be written as linear combination of atmost R columns of H.





DEFINITION

DEFINITION

Let S be subset of PG(N,q). The set S is called ρ-saturatingwhen every point P from PG(N,q) can be written as linearcombination of at most ρ+ 1 points of S.

Covering radius ρ for linear [n, k ,d ]-codeequivalent with

(ρ− 1)-saturating set in PG(n − k − 1,q)





1-SATURATING SETS





2-SATURATING SETS





1-SATURATING SET IN PG(3,q) OF SIZE 2q + 2





EXAMPLE OF ÖSTERGÅRD AND DAVYDOV

Let Fq = {a1 = 0,a2, . . . ,aq}.

H1 =

1 · · · 1 0 0 0 · · · 0a1 · · · aq 1 0 0 · · · 0a2

1 · · · a2q 0 0 1 · · · 1

0 · · · 0 0 1 a2 · · · aq

Columns of H1 define 1-saturating set of size 2q + 1 inPG(3,q).






H2 =

1 · · · 1 0 0 · · · 0 0 · · · 0 0a1 · · · aq 1 0 · · · 0 0 · · · 0 0a2

1 · · · a2q 0 1 · · · 1 0 · · · 0 0

0 · · · 0 0 a2 · · · aq a21 · · · a2

q 00 · · · 0 0 0 · · · 0 a1 · · · aq 10 · · · 0 0 0 · · · 0 1 · · · 1 0

,

Columns of H2 define 2-saturating set of size 3q + 1 inPG(5,q).





GENERALIZATION OF RESULTS OF ÖSTERGÅRD AND

DAVYDOV

Generalization of results of Östergård and Davydov by LinsDenaux.L. Denaux, Constructing saturating sets in projectivespaces using subgeometries. (arXiv:2008.13459)





The case PG(2, p), p primeDual code of PG(2, q)

OUTLINE











CODES FROM PROJECTIVE PLANES

PG(2,q), q = ph, p prime, h ≥ 1.Points Pj , j = 1, . . . ,q2 + q + 1, and lines ì ,i = 1, . . . ,q2 + q + 1.Incidence matrix

G =

← lines ì

↑points Pj

withGij = 1 iff Pj ∈ ì ,Gij = 0 iff Pj 6∈ ì .






CODE DEFINED BY THE INCIDENCE MATRIX

G = incidence matrix of PG(2,q), q = ph, p prime, h ≥ 1.G = generator matrix of [n, k ,d ]-code C = C(2,q) over Fp,withn = q2 + q + 1,

k =

(p + 1

2

)h

+ 1,

d = q + 1.






MINIMUM WEIGHT

THEOREM (ASSMUS AND KEY)

If w(c) = q + 1, then c = incidence vector of a line, up to ascalar multiple.

Question: What is the second smallest weight?Example: `1 − `2 = codeword of weight 2q.






THEOREM (CHOUINARD)

The second smallest weight of C(2,p), p prime, is 2p and allcodewords of weight 2p are, up to a scalar multiple, thedifference of two lines.

THEOREM (FACK, FANCSALI, STORME, VAN DE VOORDE,WINNE)

The non-zero codewords of weight at most 2p + (p − 1)/2 inC(2,p), p prime, p ≥ 11, are:

The scalar multiples of the lines (weight = p + 1).The scalar multiples of the difference of two lines:ρ(`1 − `2) (weight = 2p).A linear combination of two lines ρ1`1 + ρ2`2, ρ1 + ρ2 6= 0(weight = 2p + 1).






THEOREM (BAGCHI)Let p > 5 be prime.Then the fourth smallest non-zero weight of C(2,p) is 3p − 3.The only codewords of C(2,p) of Hamming weight smaller than3p − 3 are the linear combinations of at most two lines inPG(2,p).






A SPECIAL CODEWORD FOR p ODD PRIME

Let p be odd prime.Take three concurrent lines m1, m2 and m3 in PG(2,p).Choose coordinates (X0,X1,X2) such that m1 : X0 = 0,m2 : X1 = 0, and m3 : X0 = X1, respectively.Choose for each line mi some scalar µi ∈ Fp.Define c ∈ C(2,p) as follows:

c(P) =

λ1 + µ1 if P = (0,1, λ1),λ2 + µ2 if P = (1,0, λ2),−λ3 + µ3 if P = (1,1, λ3),µ1 + µ2 + µ3 if P = (0,0,1),0 otherwise.

(w(c) = 3p−3 if µ1 +µ2 +µ3 = 0 and w(c) = 3p−2 otherwise)Leo Storme Linear codes and substructures of finite geometry





SMALL WEIGHT CODEWORDS IN C(2,q)

THEOREM (SZONYI, WEINER)

Let c be a codeword of C(2,p), p > 17 prime. Ifw(c) 6 max{3p + 1,4p − 22}, then c is either a linearcombination of at most three lines or a special codeword.

THEOREM (SZONYI, WEINER)

Let c be a codeword of C(2,q), with 27 < q, q = ph, p prime. Ifw(c) < (b√qc+ 1)(q + 1− b√qc), when 2 < h, or

w(c) < (p−1)(p−4)(p2+1)2p−1 , when h = 2,

then c is a linear combination of exactly⌈w(c)

q+1

⌉different lines.






SMALL WEIGHT CODEWORDS IN Cn−1(n,q)

THEOREM

Let q = ph with p prime.1 The non-zero codewords of Cn−1(n,q) of minimum weight

are scalar multiples of hyperplanes, and have weight θn−1.2 There are no codewords of Cn−1(n,q) with weight in the

interval ]θn−1,2qn−1[.3 The codewords of weight 2qn−1 in Cn−1(n,q) are the

scalar multiples of the difference of the incidence vectorsof two distinct hyperplanes of PG(n,q).

1. Assmus and Key.2. Lavrauw, Storme, Sziklai, Van de Voorde.3. Polverino, Zullo.






SMALL WEIGHT CODEWORDS IN Cn−1(n,q)

THEOREM (ADRIAENSEN, DENAUX, STORME, WEINER)

Let c be a codeword of Cn−1(n,q), with n > 3, andw(c) 6

(4q −O(

√q))qn−2.

Then c is a linear combination of incidence vectors ofhyperplanes through a common (n − 3)-space.






SMALL WEIGHT CODEWORDS OF THE CODES Cs,t(n,q)

DEFINITION

Let s < t .

Consider an ordering Π(s)i , i = 1, . . . ,

[n + 1s + 1

]q, of the

s-spaces of PG(n,q) and an ordering Π(t)j ,

j = 1, . . . ,[

n + 1t + 1

]q, of the t-spaces of PG(n,q).







DEFINITION

(1) The incidence matrix Gs,t , s < t , of the s-spaces andt-spaces of PG(n,q) is the matrix whose rows correspond tothe t-spaces and whose columns correspond to the s-spaces ofPG(n,q), and where

(Gs,t )i,j =

{1⇔ Π

(s)j ⊆ Π

(t)i ,

0⇔ Π(s)j 6⊆ Π

(t)i .

(2) Let q = ph, p prime, h ≥ 1, and let s < t . The codeCs,t (n,q) is the linear code over the prime field Fp generated bythe incidence matrix Gs,t of PG(n,q).







DEFINITION

Let ∆s,t , s < t , denote the incidence system whose points andblocks are the s-spaces and t-spaces in PG(n,q), respectively,and the incidence is inclusion. This means that a block consistsof all the s-spaces of PG(n,q) contained in a fixed t-space ofPG(n,q).

THEOREM (BAGCHI AND INAMDAR)

The minimum weight of Cs,t (n,q) is[

t + 1s + 1

]q, and the

minimum weight vectors are the scalar multiples of theincidence vectors of the blocks of ∆s,t .







THEOREM (ADRIAENSEN, DENAUX)

All codewords of Cs,t (n,q) up to weight(3−O

(1q

))[t + 1s + 1

]q

are linear combinations at most two blocks of ∆s,t .






NECESSARY CONDITION FOR DUAL CODEWORD

PG(2,q), q = ph, p prime, h ≥ 1.Points Pj , j = 1, . . . ,q2 + q + 1, and lines ì ,i = 1, . . . ,q2 + q + 1.Incidence matrix

G =

=

g1...

gq2+q+1

← lines ì

↑points Pj

withGij = 1 iff Pj ∈ ì ,Gij = 0 iff Pj 6∈ ì .






A NECESSARY GEOMETRIC CONDITION

c ∈ C(2,q)⊥ iff c · gi = 0.If c ∈ C(2,q)⊥, then supp(c) defines a set of points ofPG(2,q) intersecting every line in zero or at least twopoints.






KNOWN RESULTS

THEOREM

The minimum weight of C(2,p)⊥, p prime, is 2p.The minimum weight of C(2,2h)⊥ is 2h + 2, and everycodeword in C(2,2h)⊥ of weight 2h + 2 is the incidencevector of a hyperoval of PG(2,2h).The minimum weight of C(2,ph)⊥, p prime, p > 7, h > 1,satisfies

12q + 187

≤ d ≤ 2q + 1− (q − 1)/(p − 1).

Upper bound: line minus Rédei-type blocking set of sizeq + (q − 1)/(p − 1).






KNOWN RESULTS

THEOREM

The minimum weight of C(2,2h)⊥ is 2h + 2, and everycodeword in C(2,2h)⊥ of weight 2h + 2 is the incidencevector of a hyperoval of PG(2,2h).The minimum weight of C0,k (n,q)⊥, q even, is(q + 2)qn−k−1.






KNOWN RESULTS

THEOREM (ADRIAENSEN, DENAUX)

d(Cs,t (n,q)⊥) = d(C0,1(n − t + 1,q)⊥).

If p is prime, then the minimum weight codewords ofCs,t (n,p)⊥ are the scalar multiples of the standard words,and thus have weight 2pn−t .






REFERENCES

I. Landjev, Linear codes over finite fields and finiteprojective geometries. Discr. Math. 213 (2000), 211-244.I. Landjev and L. Storme, Galois geometries and codingtheory. Chapter in Current research topics in Galoisgeometry (J. De Beule and L. Storme, Eds.), NOVAAcademic Publishers (2012), 187-214.






Thank you very much for your attention!


Linear codes and substructures of finite geometry

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