International audienceWe can associate to each linear code C defined over a finite field the matroid M[H] of its parity check matrix H. For any matroid M one can define its generalized Hamming weights which are the same as those of the code C. In [2] the authors show that the generalized Hamming weights of a matroid are determined by the N-graded Betti numbers of the Stanley-Reisner ring of the simplicial complex whose faces are the independent set of M. In this talk we go a step further. Our practical results indicate that the generalized Hamming weights of a linear code C can be obtained from the monomial ideal associated with a test-set for C. Moreover, recall that in [3] we use the Gröbner representation of a linear code C to provide a test-set for C. Our results are still a work in progress, but its applications to Coding Theory and Cryptography are of great value

Martínez-Moro, Edgar

Márquez-Corbella, Irene

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HAL Id: hal-01409298https://hal.inria.fr/hal-01409298Submitted on 5 Dec 2016HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Betti Numbers and Generalized Hamming WeightsIrene Márquez-Corbella, Edgar Martínez-MoroTo cite this version:Irene Márquez-Corbella, Edgar Martínez-Moro. Betti Numbers and Generalized Hamming Weights.22nd Conference on Applications of Computer Algebra (ACA 2016), Aug 2016, Kassel, Germany.￿hal-01409298￿Betti Numbers and Generalized Hamming WeightsIrene Márquez-Corbella and Edgar Martínez-MoroDept. of Mathematics, Statistics and O. Research, University of La Laguna, Spain.irene.marquez.corbella@ull.esInstitute of Mathematics, University of Valladolid, Spain. Edgar.Martinez@uva.esWe can associate to each linear code C defined over a finite field the matroidM[H] of its parity check matrix H. For any matroid M one can define its gener-alized Hamming weights which are the same as those of the code C . In [2] theauthors show that the generalized Hamming weights of a matroid are determinedby the N-graded Betti numbers of the Stanley-Reisner ring of the simplicial com-plex whose faces are the independent set of M. In this talk we go a step further.Our practical results indicate that the generalized Hamming weights of a linearcode C can be obtained from the monomial ideal associated with a test-set for C .Moreover, recall that in [3] we use the Gröbner representation of a linear code Cto provide a test-set for C .Our results are still a work in progress, but its applications to Coding Theoryand Cryptography are of great value.1 Notation and PrerequisitesWe begin with an introduction of basic definitions and some known results. By N,Z, Fq (where q is a primer power) we denote the set of positive integers, the set ofintegers and the finite field with q elements, respectively.Definition 1 A matroid M is a pair (E, I) consisting of a finite set E called groundset and a collection I of subsets of E called independent sets, satisfying the follow-ing conditions:1. The empty set is independent, i.e. /0 ∈ I2. If A ∈ I and B⊂ A, then B ∈ I3. If A,B ∈ I and |A|< |B|, then there exists e ∈ B\A such that A∪{e} ∈ ILet M = (E, I) be a matroid. A maximal independent subset of E is calleda basis of M. A direct consequence of the previous definition is that all basesof M have the same cardinality. Thus, we define the rank of the matroid M as thecardinality of any basis of M, denoted by rank(M). A subset E that does not belong1to I is called dependent set. Minimal dependent subsets of E are known as circuitsof M. A set is said to be a cycle if it is a disjoint union of circuits. The collectionof cycles of M is denoted by C (M). For all σ ∈ E, the nulity function of σ is givenby n(σ) := |σ |− rank(Mσ ) with rank(Mσ ) = max{|A| | A ∈ I and A⊂ σ}, i.e. therestriction of rank(M) to the subsets of σ .Let us consider an m× n matrix A in Fq whose columns are indexed by E ={1, . . . ,n} and take I to be the collection of subsets J of E for which the columnvectors{A j | j ∈ J}are linearly independent over Fq. Then (E, I) defines a matroiddenoted by M[A]. A matroid M = (E, I) is Fq-representable if it is isomorphic toM[A] for some A ∈ Fm×nq . Then the matrix A is called the representation matrix ofM. The following well known results describes the relation between the collecitonof all cycles of a matroid M and its representation matrix.Proposition 1 Let M = (E, I) be a Fq-representable matroid. Then C (M) is thenull space of a representation matrix of M. Furthermore, the dimension of C (M)is |E|− rank(M).Let ∆ be a simplicial complex on the finite ground set E. Let K be a field andlet x be the indeterminates x = {xe | e ∈ E}. The Stanley-Reisner ideal of ∆ is, bydefinition,I∆ = 〈xσ | σ /∈ ∆〉The Stanley-Reisner ring of I∆, denoted by R∆, is defined to be the quotientring R∆ =K[x]I∆. This ring has a minimal free resolution as NE-graded module:0 ←− R∆ ←− P0 ←− P1 ←− ·· · ←− Pl ←− 0where each Pi is given by Pi =⊕α∈NE K[x](−α)βi,α . We write βi,α for theNE-graded Betti Numbers of ∆.1.1 Matroids and Simplicial complexA matroid M = (E, I) is a simplicial complex whose faces are the independentsets. Thus, IM := 〈xσ | σ ∈ C 〉 where C is the set of all circuits of M. DefineNi = {σ ∈ N | n(σ) = d}.Theorem 1 ([2]Theorem 1) Let M be a matroid on the ground set E. Let σ ⊂ E.Then, βi,σ 6= 0 if and only if σ is minimal in Ni.Definition 2 Let M = (E, I) be a matroid, we define the generalized Hammingweights of M to be di = min{|σ | | n(σ) = i}.Corollary 1 Let M be a matroid on the ground set E. Then,di = min{d | βi,d 6= 0 for all 1≤ i≤ |E|− rank(M)} .21.2 Matroids and linear codesAn [n,k]q linear code C is a k-dimensional subspace of Fnq. We define a generatormatrix of C to be a k×n matrix G whose row vectors span C , while a parity checkmatrix of C is an (n− k)×n matrix H whose null space is C .Let us denote by dH(·, ·) and wH(·) the Hamming distance and the Hammingweight on Fnq, respectively. We write d for the minimum Hamming distance ofthe code C , which is equal to its minimum weight. Thus, the error correctingcapability of C is t =⌊d−12⌋where b·c is the greatest integer function. For everycodeword c ∈ C its support, supp(c), is defined as its support as a vector in Fnq, i.e.supp(c) = {i | ci 6= 0}. We will denote by MC the set of codewords of minimalsupport of C .A test-set TC for C is a set of codewords such that for every word y ∈ Fnq,either y belongs to the set of coset leaders, or there exists an element t ∈ TC suchthat wH(y− t)< wH(y).Definition 3 The rth generalized Hamming weight of C denoted by dr(C ) is thesmallest support of an r-dimensional subcode of C . That is,dr(C ) = min{supp(D) | D⊆ C and rank(D) = r}In [3] the authors associate a binomial ideal to an arbitrary linear code providedby the rows of a generator matrix and the relations given by the additive table ofthe defining field.Let X denote n vector variables X1, . . . ,Xn such that each variable Xi can bedecomposed into q−1 components xi,1, . . . ,xi,q−1 with i = 1, . . . ,n. A monomial inX is a product of the form:Xu = Xu11 · · ·Xunn =(xu1,11,1 · · ·xu1,q−11,q−1)· · ·(xun,1n,1 · · ·xun,q−1n,q−1)where u ∈ Zn(q−1)≥0 . The total degree of Xu is the sum deg(Xu) = ∑ni=1 ∑q−1j=1 ui, j.When u = (0, . . . ,0), note that Xu = 1. Then, the polynomial ring K[X] is the setof all polynomials in X with coefficients in K.Recall that the multiplicative group F∗q of nonzero elements of Fq is cyclic.A generator of the cyclic group F∗q is called a primitive element of Fq, i.e. Fqconsist of 0 and all powers from 1 to q− 1 of that primitive element. Let α bea primitive element of Fq. We define by RXi , the set of all the binomials on thevariables Xi associated to the relations given by the additive table of the field Fq =〈α j | j = 1, . . . ,q−1〉∪{0}, i.e.RXi ={{xi,uxi,v− xi,w | αu +αv = αw} ∪ {xi,uxi,v−1 | αu +αv = 0}}3with i = 1, . . . ,n. Note that there are(q2)different binomials in RXi . We define RXas the ideal generated by the union of all binomial ideals RXi , i.e. RX =〈∪ni=1RXi〉We will use the following characteristic crossing functions. These applicationsaim at describing a one-to-one correspondence between the finite field Fq with qelements and the standard basis of Zq−1, denoted as Eq = {e1, . . . ,eq−z} where eiis the unit vector with a 1 in the i-th coordinate and 0’s elsewhere.∆ : Fq −→ Eq∪{0} ⊆ Zq−1 and ∇ : Eq∪{0} −→ Fq1. The map ∆ replaces the element a = α i ∈ Fq by the vector ei and 0 ∈ Fq bythe zero vector 0 ∈ Zq−1.2. The map ∇ recovers the element α j ∈ Fq from the unit vector e j and the zeroelement 0 ∈ Fq from the zero vector 0 ∈ Zq−1.These maps will be used with matrices and vectors acting coordinate-wise. Al-though ∆ is not a linear function. Note that we have:X∆a ·X∆b = X∆a+∆b = X∆(a+b) mod RX for all a,b ∈ Fnq.Let C be an [n,k]q linear code. We define the ideal associated to C as thebinomial ideal:I(C ) =〈{X∆a−X∆b | a−b ∈ C}〉⊆K[X]Given the rows of a generator matrix C , labelled by {w1, . . . ,wk} ⊆ Fnq, wedefine the following ideal:I+(C ) =〈 {X∆(α jwi)−1}i=1,...,nj=1,...q−1∪ {RXi}i=1,...,n〉⊆K[X]Theorem 2 [3][Theorem 2.3] I(C ) = I+(C )Remark 1 In the binary case, given a generator matrix G∈Fk×n2 of an [n,k]2-codeC and let label its rows by {w1, . . . ,wk} ⊆ Fn2. We define the ideal associated to Cas the binomial ideal:I+(C ) =〈{Xwi−1}i=1,...,k ∪{x2i −1}i=1,...,n〉⊆K[X]Now, let G = {g1, . . . ,gs} be the reduced Gröbner basis of the ideal I+(C ) withrespect to , where we take  to be any degree compatible ordering on K[X] with4X1 ≺ . . . ≺ Xn. By Lemma [3][Lemma 3.3] we know that all elements of G \RXare in standard form, so for gi ∈ G \RX with i = 1, . . . ,s, we definegi = X∆g+i −X∆g−i with X∆g+i  X∆g−i and g+i −g−i ∈ C .Using [3][Proposition 4], we know that the set T ={g+i −g−i | i = 1, . . . ,s}isa test-set for C .Example 1 Consider the [6,3,2]2 binary code C defined by the following genera-tor matrix:G = 1 0 0 0 0 10 1 1 0 1 00 0 0 1 1 1 ∈ F3×62Let us label the rows of G by w1 and w2. By the previous theorem, the idealassociated to the linear code C may be defined as the following ideal:I+(C ) =〈{Xwi−1}i=1,2 ∪ {RXi}i=1,...,6〉=〈 x1x6−1x2x3x5−1x4x5x6−1 ∪ {x2i −1}i=1,...,6〉If we compute a reduced Gröbner basis G of I+(C ) we obtained a test-set consist-ing of 4 codewords:TC = {(1,0,0,0,0,1),(0,1,1,0,1,0),(0,1,1,1,1,0,1),(0,0,0,1,1,1)}For fuller discussion of this algebraic structure see [4, 1] and the references therein.The connection between linear codes and matroids will turn out to be funda-mental for the development of the subsequent results. Thus, a brief review will beprovided here.Given an m× n matrix H in Fq, then H can be seen not only as the repre-sentation matrix of the Fq-representable matroid M[H] but also as a parity checkmatrix of an [n,k]-code C . Furthermore, there exists a one to one correspondencebetween Fq-representable matroids and linear codes, since for any H,H ′ ∈ Fm×nq ,M[H] = M[H ′] if an only if H and H ′ are parity check matrices of the same codeC . This association enables us to work with Fq-representable matroids and linearcodes as if they were the same object and thus we can conclude some properties oflinear codes using tools from matroid theory and vice-versa.52 Our ConjectureLet M = (E, I) be a matroid and C be the set of all circuits of M. Consider Ta collection of cycles of M with the following property:⋃τ∈C τ =⋃τ∈T τ . Wedefine the ideal IT = 〈xσ | σ ∈T 〉.Conjecture 1 Let β ′i,α the NE-graded betti number of IT , related with the minimalfree resolution of R = K[X ]IT as NE-graded module. Then, we have a similar resultas Theorem 1 and Corollary 1.If we talk about linear codes, the conjecture allows us to compute the set ofgeneralized Hamming weight of a linear code C using a Test-set for C , in otherwords, by computing a Grobner basis of the ideal associated to C .Corollary 2 Let TC be a test-set for the linear code C . Consider the monomialideal: ITC = 〈xσ | σ ∈TC 〉. Let β ′i,α the NE-graded betti numbers of ITC . Then,di(C ) = min{d | β ′i,d 6= 0}for 1≤ i≤ n− kExample 2 Now we use the same code of Example 1. In this case the supportof a test-set TC is given by: T = {{2,3,5},{2,3,4,6},{4,5,6},{1,6}} i.e. weconsider the ideal: IT = 〈x2x3x5,x2x3x4x6,x4x5x6,x1x6〉 ⊆ K[x1, . . . ,x6]. We getthe Betti diagram1 2 31 12 2 13 1 4 2Thus β ′1,2, β′2,4 and β′3,6 are the minimal β′i,d 6= 0 with i = 1,2,3. Or equivalently,d1 = 2, d2 = 4 and d3 = 6.References[1] M. Borges-Quintana, M.A. Borges-Trenard, P. FitzPatrick and E. Martínez-Moro. Gröbnerbases and combinatorics for binary codes. Applicable Algebra in Engineering, Communica-tion and Computing. 19(5): 393-411, 2008.[2] J. T. Johnsen and H. Verdure. Hamming weights and Betti numbers of Stanley–Reisner ringsassociated to matroids. Applicable Algebra in Engineering, Communication and Computing.24(1): 73-93, 2013.[3] I. Márquez-Corbella, E. Martínez-Moro and E. Suárez-Canedo. On the ideal associated to alinear code. Advances in Mathematics of Communications (AMC). 10(2): 229-254, 2016.[4] I. Márquez-Corbella and E. Martínez-Moro. Algebraic structure of the minimal support code-words set of some linear codes. Advances in Mathematics of Communications (AMC). 5(2):233-244, 2011.6

Betti Numbers and Generalized Hamming Weights

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Betti Numbers and Generalized Hamming Weights

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