International audienceWe will show how one can compute all reduced Gröbner bases with re-spect to a degree compatible ordering for code ideals -even though these binomial ideals are not toric. To this end, the correspondence of linear codes and binomial ideals will be briefly described as well as their resemblance to toric ideals. Finally, we will hint at applications of the degree compatible Gröbner fan to the code equivalence problem

Dück, Natalia

Martínez-Moro, Edgar

Márquez-Corbella, Irene

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HAL Id: hal-01088432https://hal.archives-ouvertes.fr/hal-01088432Submitted on 28 Nov 2014HAL 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.On the fan associated to a linear codeNatalia Dück, Irene Márquez-Corbella, Edgar Martínez-MoroTo cite this version:Natalia Dück, Irene Márquez-Corbella, Edgar Martínez-Moro. On the fan associated to a linear code. 4th ICMCTA - Fourth International Castle Meeting on Coding Theory and Applications, Sep 2014,Palmela, Portugal. ￿hal-01088432￿On the fan associated to a linear code∗Natalia Dück, Irene Márquez-Corbella and Edgar Martı́nez-MoroAbstract We will show how one can compute all reduced Gröbner bases with re-spect to a degree compatible ordering for code ideals - even though these binomialideals are not toric. To this end, the correspondence of linear codes and binomialideals will be briefly described as well as their resemblance to toric ideals. Finally,we will hint at applications of the degree compatible Gröbner fan to the code equiv-alence problem.Key words: Linear code, Gröbner basis, Gröbner fan1 IntroductionThe Gröbner fan of an ideal in the commutative polynomial ring consists of polyhe-dral cones indexing the different leading ideals and is thus the geometric collectionof all reduced Gröbner bases for this ideal. One application of the Gröbner fan is theso-called Gröbner walk which is the conversion of Gröbner bases.With the software system TiGERS in [5] (Toric Gröbner bases Enumeration byReverse Search) an efficient alternative for computing the Gröbner fan has beenprovided for the special case of toric ideals. Indeed, by identifying a reverse searchNatalia DückHamburg University of Technology, Germany e-mail: natalia.dueck@tuhh.deIrene Márquez-CorbellaINRIA Saclay- École Polytechnique, France e-mail: irene.marquez-corbella@inria.frEdgar Martı́nez-MoroInstitute of Mathematics, University of Valladolid, Spain e-mail: edgar@maf.uva.es∗ First author is partially supported by a grant of the Deutscher Akademischer Austauschdienst.Second and third authors are supported by the Spanish MINECO grant MTM2012-36917-C03-03.12 Natalia Dück, Irene Márquez-Corbella and Edgar Martı́nez-Morotree on the cones of the Gröbner fan, a memory-less combinatorial Gröbner walkcan be established that furthermore, requires no cost weight vectors.Linear codes, on the other hand, can be linked to this whole subject by associ-ating to each linear code a binomial ideal that encodes the information about thecode in the exponents. This correspondence proved to be very beneficial as it pro-vided new approaches to several well-known problems in coding theory. Almostall applications, however, require the computation of a degree compatible Gröbnerbasis.In this work, it will be shown how methods from the software system TiGERSdeveloped by Rekha R. Thomas (see [5]) can be modified in order to compute allreduced Gröbner bases with respect to a degree compatible ordering for code ideals- even though these binomial ideals are not toric. To this end, the correspondenceof linear codes and binomial ideals will be briefly described as well as their resem-blance to toric ideals. Finally, we will hint at applications of the degree compatibleGröbner fan to the code equivalence problem.2 The degree compatible Gröbner fanIn this work we shall use the notion of Gröbner basis and the ideal associated to alinear code. Due to the restriction of the space we will not define what a Gröbnerbasis is, the reader can find a good introductory text for example in [3]. Also for sim-plicity we will restrict ourselves to binary linear codes even if all the computationcould be done in general (see [7] for the ideal associated to a q-ary linear code).Let K[x] be the polynomial ring with variables x = x1, . . . ,xn and coefficients anarbitrary field K. We will define the ideal associated to a binary linear code C oflength n asI = I(C ) = 〈{x∆a−x∆b | a−b ∈ C }〉 ⊆K[x],where the operation ∆ means substitute the 0̄, 1̄ elements in the binary field F2 bythe corresponding 0,1 in the set of integers Z. In this extended abstract the ∆ willbe omitted if no confusion arises to simplify the notation.This binomial ideal has been proved valuable for several applications and cap-tures the combinatorial properties of the code (see [6] and the references therein).Note that for those applications K can be the binary field, which is the usual election,and in this case we must explicitly mark which terms are the leading terms.In this paper it shall be assumed that the leading term of a binomial is the onewith coefficient 1 and the non leading term has coefficient -1. Abusing the notationif K is the binary field, since 1≡−1, this writing of the binomials will be assumedas a formal pointer (in [5] the leading terms were underlined).Note also that the explicit knowledge of the underlying term order is not neces-sary. In fact, in all the following computations only the leading term of each bino-mial has to be known.On the fan associated to a linear code 3In the rest of the paper we will use the following notation and concepts from [5]:• G(I) is the reduced Gröbner basis for the ideal I w.r.t. the monomial order ,• C(I) is the Gröbner cone corresponding to G(I).• T(I) is the reverse search tree for the ideal I as constructed in [5, Definition 2.5]Note that in [5] the complete Gröbner fan is considered, i.e., the whole Rn, sincethe considered toric ideals are homogeneous w.r.t. a certain grading. This is not thecase for our code ideal and so here the Gröbner fan is considered only in Rn+.Proposition 1. [4] Let  be a term order and v ∈C(I). For any u ∈ Rn holdsltu(I) = ltv(I) ⇐⇒ ltu(g) = ltv(g) ∀g ∈ G(I),where ltu stands for the leading (initial) term (ideal) induced by the order  givenby the weight vector u.Note that it is a well known fact that a reduced Gröbner basis for an ideal Iw.r.t. a certain monomial order is degree compatible if and only if the correspondingGröbner cone contains the all-one vector 1. From a coding-theory point of view,degree compatible orderings are the ones one must analyze since the weight of avector is translated on the degree of a monomial. In this sense degree compatibleorderings provide us a test set for the code and therefore a gradient descent decodingalgorithm, see [1]. The following proposition characterizes when there is a uniquedegree compatible Gröbner basis.Proposition 2. Let G be a reduced Gröbner basis for I(C ) w.r.t. a certain degreecompatible ordering. The Gröbner basis G is the only reduced degree compatibleGröbner basis for I(C ) if and only ifdeg(xa)> deg(xb) for all xa−xb ∈ G . (1)Proof. Assume that (1) holds but there is another Gröbner basis G ′ for I(C ) w.r.t.another degree compatible order′. Since′ is degree compatible, lt′(g) = lt(g)for all g ∈ G . And by Proposition 1 we see that lt′(I(C )) = lt(I(C )) and thus,G = G ′.Or equivalently, we can argue that the all-one vector is in the interior of the coneC(I(C )) and so clearly it cannot be contained in another cone in the Gröbner fan.In order to show the other direction assume that (1) does not hold, i.e., thereis at least one binomial xa − xb in G such that deg(xa) = deg(xb). Then 1 /∈Int(C(I(C ))) and in particular, there must be a neighbouring cone that also con-tains 1 and thus corresponds to a degree compatible ordering. utIn terms of the Gröbner fan the above proposition can also be expressed as fol-lows: A reduced Gröbner basis G w.r.t. a degree compatible ordering is the onlydegree compatible Gröbner basis if and only if the all-one vector 1 lies in the inte-rior of the Gröbner cone of G .4 Natalia Dück, Irene Márquez-Corbella and Edgar Martı́nez-MoroWe say that two binary linear codes C1 and C2 are permutation equivalent pro-vided there is a permutation of coordinates which sends C1 to C2. In the same fash-ion two binomial degree compatible Gröbner bases are permutation equivalent ifthere is a permutation of the variables that transforms one into the other. There isa close relationship between code equivalence and the equivalence of the degreecompatible Gröbner bases associated to their code ideals stated as follows: If thetwo degree compatible Gröbner bases are permutation equivalent so are the codes,unfortunately the converse is not true, given two permutation equivalent codes notall the degree compatible Gröbner bases are permutation equivalent (only two ofthem should be). The reader can see [2] for a proof of this discussion.Indeed if one has only a unique degree compatible Gröbner basis for a given code(Proposition 2) checking permutation equivalence is reduced to checking if the twounique bases are permutation equivalent using the techniques in [2]. If this is not thecase one needs to compute the whole set of degree compatible Gröbner bases whichwe call the degree compatible Gröbner fan. We will tackle this task in the followingsection.3 Adapting the TiGERS strategyWe can adapt the TiGERS Algorithm in [5] for computing the degree compatibleGröbner fan for I(C ) as follows: We start with a degree compatible Gröbner basis(note that this basis can be computed by the algorithm stated in [2]). By Proposi-tion 2 we can determine whether it is the only degree compatible Gröbner basis ornot. If not, we flip only those facet binomials where both terms have the same degreeand recompute the Gröbner basis. Unfortunately due the lack of space these stepscan not be detailed in this extended abstract but they are showed in [5]. Lemma 1below guarantees that we will always find at least one facet binomial where bothterms have the same degree. Additionally, we can employ the reverse search treedefined in [5] for traversing the Gröbner cones that are degree compatible.Lemma 1. Let G be the reduced Gröbner basis for I(C ) w.r.t. a degree com-patible ordering. If G is not the only degree compatible Gröbner basis, that is1 /∈ Int(C(I(C ))), then among all the facet binomials of G is at least one binomialxα −xβ such that deg(xα) = deg(xβ ).Proof. Let G ={xαi −xβi | 1≤ i≤ j+ k}and order the binomials such that deg(xαi)>deg(xβi) for 1≤ i≤ j and deg(xαi) = deg(xβi) for 1+ j ≤ i≤ j+ k.Assume that all facet binomials are such that the degree of the leading term isstrictly greater than the degree of the other term. Then the coneC′ ={u ∈ Rn+ | αi ·u≥ βi ·u for all 1≤ i≤ j}equals the Gröbner cone C(I(C )) of the Gröbner basis G . But then 1 ∈ Int(C′) =Int(C(I(C ))), which is a contradiction. utOn the fan associated to a linear code 5Lemma 2. Let Gnew be the reduced Gröbner basis obtained from Gold by flippingthe facet binomial xα − xβ . Any new leading terms in Gnew, i.e., leading terms ofGnew that do not appear in Gold , are divisible by xα .Proof. Any new leading terms arise from the Gröbner basis computation of thequasi-monomial idealT :={xβ −xα}∪T ′, T ′ :={xαi | xαi −xβi ∈ Gold}that consists of the designated flipping binomial with changed leading term and allthe other leading terms in Gold . To be more precise, a new leading term arises froman S-polynomial of the formS(xβ −xα ,xαi)= xγ xα , where xγ = lcm(xβ ,xαi)/xβ ,which is not being reduced to zero by the elements in T . When computing a Gröbnerbasis, then this S-polynomial is either reduced to zero or its remainder on division bythe set T is added to the Gröbner basis of T . We distinguish the following situations:1. Neither xβ nor any monomial in T ′ divides xγ xα : The monomial xγ xα cannot befurther reduced and thus is being attached to the Gröbner basis of T .2. A monomial in T ′ divides xγ xα : The monomial xγ xα is being reduced to zeroand thus, this S-polynomial results in no new term.3. The monomial xβ divides xγ xα : Since xα and xβ have disjoint support (see [1]),xβ has to divide xγ , the monomial xγ xα is reduced toxγ xα −xα xγxβ(xβ −xα)=xγxβ(xα)2 .So, whenever the S-polynomial cannot be reduced to zero, we obtain a monomialwhich is divisible by xα . utProposition 3. T(I(C )) is an acyclic directed graph with a unique sink that wewill call the reverse search tree.Proof. We prove that T(I(C )) is a tree by showing that there is no cycle in thisconstruction. We show this by contradiction. For the other claims see the proof of [5,Theorem 2.6].Assume that there is a cycle in the reverse search tree, say G1 −→ G2 −→ . . .−→G` −→ G1, where Gi+1 is obtained from Gi by flipping along xαi−xβi . Then Gi con-tains this binomial with leading term xαi and Gi+1 with leading term xβi . Inspectingthe cycle we see that the binomial xα1−xβ1 lies in G1 with leading term xα1 and ap-pears in G2 with leading term xβ1 . Then no binomial in G2 has the leading term xα1 .However, as we arrive at G1 after ` flipping steps, we conclude that xα1 −xβ1 mustbe inserted at some successive flipping step. Assume that this happens in the i1thflipping process, 1 < i1 ≤ `. Then by Lem. 2, xα1 is divisible by xαi1 . And since G1is a Gröbner basis this implies that xαi1 cannot be the leading term of any element in6 Natalia Dück, Irene Márquez-Corbella and Edgar Martı́nez-MoroG1; it must have been inserted as a new leading term during some preceding flippingstep, say i2 < i1. By the same argument the monomial xαi1 is divisible by xαi2 andthen xαi2 cannot appear as the leading term of any element in G1. Continuing thisprocess we get a decreasing sequence of indices i1 > i2 > i3 > .. . which eventuallymust terminate, say after k steps, i.e., ik = 1. Then xαik = xα1 and from the divisi-bility relations xαik | xαik−1 | . . . | xαi2 | xαi1 | xα1 we actually obtain equality of allleading terms of the flipping binomials. However, this is a contradiction. utThe following proposition states the discussion at the end of Section 2.Proposition 4. Two linear codes C1 and C2 are permutation-equivalent if and onlyif they have the same degree compatible Gröbner fan structure, i.e., there is per-mutation σ ∈ Sn such that σ (G f an(C1)) = G f an(C2), where σ (G f an(C1)) meanspermuting the variables in each of the degree compatible Gröbner basis within thefan.Example 1. Consider two binary [6,3] codes C1 and C2 with respective parity checkmatricesH1 =1 1 0 0 0 00 0 1 1 0 00 0 0 0 1 1 , H2 = 1 1 1 1 1 10 0 0 1 0 10 1 0 1 0 0In [2, Example 2 and 5] it is shown that these codes are not permutation-equivalent.Here, we show how the degree compatible Gröbner fans of both codes can be em-ployed to show their non-equivalence. The degree compatible Gröbner fan for C1consists of 8 Gröbner basis which are all of cardinality 6 (see Ex. 2). The Gröbnerbasis for C2 w.r.t. the grevlex basis is given by{x3− x5,x1− x5,x4x5− x2x6,x2x5− x4x6,x2x4− x5x6} ∪{x2i −1 | i = 2,4,5,6}and consists of 9 elements. Thus, we can already conclude that these two codescannot be permutation-equivalent.Example 2. The reverse search tree T(I(C )) for the binary [6,3] code C1 from theprevious example with  being pure lex is given in Fig. 1.And the Gröbner bases are (the flipping binomials are underlined)G1 ={x1− x2,x3− x4,x5− x6,x22−1,x24−1,x26−1}G2 ={x1− x2,x4− x3,x5− x6,x22−1,x23−1,x26−1}G3 ={x2− x1,x4− x3,x5− x6,x21−1,x23−1,x26−1}G4 ={x1− x2,x3− x4,x6− x5,x22−1,x24−1,x25−1}G5 ={x1− x2,x4− x3,x6− x5,x22−1,x23−1,x25−1}G6 ={x2− x1,x4− x3,x6− x5,x21−1,x23−1,x25−1}G7 ={x2− x1,x3− x4,x6− x5,x21−1,x24−1,x25−1}G8 ={x2− x1,x3− x4,x5− x6,x21−1,x24−1,x26−1}.On the fan associated to a linear code 7G1G2 G4 G8G3 G5 G7G6Fig. 1 The reverse search tree for C1ConclusionsWe have shown how the computation of the degree compatible Gröbner fan of a codeis useful for determining the code equivalence problem. Anyway one can not forgetthat this is an NP-problem and therefore the Gröbner basis computation comprises ahard step. Further research in the topic points toward analyzing heuristic techniquesfor eliminating the need of transverse the whole fan or at least for trying to deducethe answer from partial information about the initial Gröbner basis.References1. Borges-Quintana, M., Borges-Trenard, M.A., Fitzpatrick, P., Martı́nez-Moro, E.: Gröbner basesand combinatorics for binary codes. Appl. Algebra Engrg. Comm. Comput. 19(5), 393–411(2008)2. Borges-Quintana, M., Borges-Trenard, M.A., Martı́nez-Moro, E.: On a Gröbner bases structureassociated to linear codes. J. Discrete Math. Sci. Cryptogr. 10(2), 151–191 (2007)3. Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms: An introduction to compu-tational algebraic geometry and commutative algebra. Undergraduate Texts in Mathematics.Springer-Verlag, New York, second edn. (1997)4. Fukuda, K., Jensen, A.N., Thomas, R.R.: Computing Gröbner fans. Math. Comp. 76(260),2189–2212 (electronic) (2007)5. Huber, B., Thomas, R.R.: Computing Gröbner fans of toric ideals. Experiment. Math. 9(3),321–331 (2000)6. Márquez-Corbella, I., Martı́nez-Moro, E.: Algebraic structure of the minimal support code-words set of some linear codes. Adv. Math. Commun. 5(2), 233–244 (2011)7. Márquez-Corbella, I., Martı́nez-Moro, E., Suárez Canedo, E.: On the ideal associated to anylinear code. Adv. Math. Commun. -(-), Submitted (2014)

On the fan associated to a linear code

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On the fan associated to a linear codeNatalia Du¨ck, Irene Ma´rquez-Corbella, Edgar Mart´ınez-MoroTo cite this version:Natalia Du¨ck, Irene Ma´rquez-Corbella, Edgar Mart´ınez-Moro. On the fan associated to a linearcode . 4th ICMCTA - Fourth International Castle Meeting on Coding Theory and Applications,Sep 2014, Palmela, Portugal. CIM Series in Mathematical Sciences (CIM-MS) to be publishedby Springer-Verlag. <hal-01088432>HAL Id: hal-01088432https://hal.archives-ouvertes.fr/hal-01088432Submitted on 28 Nov 2014HAL 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, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.On the fan associated to a linear code∗Natalia Du¨ck, Irene Ma´rquez-Corbella and Edgar Martı´nez-MoroAbstract We will show how one can compute all reduced Gro¨bner bases with re-spect to a degree compatible ordering for code ideals - even though these binomialideals are not toric. To this end, the correspondence of linear codes and binomialideals will be briefly described as well as their resemblance to toric ideals. Finally,we will hint at applications of the degree compatible Gro¨bner fan to the code equiv-alence problem.Key words: Linear code, Gro¨bner basis, Gro¨bner fan1 IntroductionThe Gro¨bner fan of an ideal in the commutative polynomial ring consists of polyhe-dral cones indexing the different leading ideals and is thus the geometric collectionof all reduced Gro¨bner bases for this ideal. One application of the Gro¨bner fan is theso-called Gro¨bner walk which is the conversion of Gro¨bner bases.With the software system TiGERS in [5] (Toric Gro¨bner bases Enumeration byReverse Search) an efficient alternative for computing the Gro¨bner fan has beenprovided for the special case of toric ideals. Indeed, by identifying a reverse searchNatalia Du¨ckHamburg University of Technology, Germany e-mail: natalia.dueck@tuhh.deIrene Ma´rquez-CorbellaINRIA Saclay- E´cole Polytechnique, France e-mail: irene.marquez-corbella@inria.frEdgar Martı´nez-MoroInstitute of Mathematics, University of Valladolid, Spain e-mail: edgar@maf.uva.es∗ First author is partially supported by a grant of the Deutscher Akademischer Austauschdienst.Second and third authors are supported by the Spanish MINECO grant MTM2012-36917-C03-03.12 Natalia Du¨ck, Irene Ma´rquez-Corbella and Edgar Martı´nez-Morotree on the cones of the Gro¨bner fan, a memory-less combinatorial Gro¨bner walkcan be established that furthermore, requires no cost weight vectors.Linear codes, on the other hand, can be linked to this whole subject by associ-ating to each linear code a binomial ideal that encodes the information about thecode in the exponents. This correspondence proved to be very beneficial as it pro-vided new approaches to several well-known problems in coding theory. Almostall applications, however, require the computation of a degree compatible Gro¨bnerbasis.In this work, it will be shown how methods from the software system TiGERSdeveloped by Rekha R. Thomas (see [5]) can be modified in order to compute allreduced Gro¨bner bases with respect to a degree compatible ordering for code ideals- even though these binomial ideals are not toric. To this end, the correspondenceof linear codes and binomial ideals will be briefly described as well as their resem-blance to toric ideals. Finally, we will hint at applications of the degree compatibleGro¨bner fan to the code equivalence problem.2 The degree compatible Gro¨bner fanIn this work we shall use the notion of Gro¨bner basis and the ideal associated to alinear code. Due to the restriction of the space we will not define what a Gro¨bnerbasis is, the reader can find a good introductory text for example in [3]. Also for sim-plicity we will restrict ourselves to binary linear codes even if all the computationcould be done in general (see [7] for the ideal associated to a q-ary linear code).Let K[x] be the polynomial ring with variables x= x1, . . . ,xn and coefficients anarbitrary field K. We will define the ideal associated to a binary linear code C oflength n asI = I(C ) = 〈{x∆a−x∆b | a−b ∈ C }〉 ⊆K[x],where the operation ∆ means substitute the 0¯, 1¯ elements in the binary field F2 bythe corresponding 0,1 in the set of integers Z. In this extended abstract the ∆ willbe omitted if no confusion arises to simplify the notation.This binomial ideal has been proved valuable for several applications and cap-tures the combinatorial properties of the code (see [6] and the references therein).Note that for those applicationsK can be the binary field, which is the usual election,and in this case we must explicitly mark which terms are the leading terms.In this paper it shall be assumed that the leading term of a binomial is the onewith coefficient 1 and the non leading term has coefficient -1. Abusing the notationif K is the binary field, since 1≡−1, this writing of the binomials will be assumedas a formal pointer (in [5] the leading terms were underlined).Note also that the explicit knowledge of the underlying term order is not neces-sary. In fact, in all the following computations only the leading term of each bino-mial has to be known.On the fan associated to a linear code 3In the rest of the paper we will use the following notation and concepts from [5]:• G(I) is the reduced Gro¨bner basis for the ideal I w.r.t. the monomial order ,• C(I) is the Gro¨bner cone corresponding to G(I).• T(I) is the reverse search tree for the ideal I as constructed in [5, Definition 2.5]Note that in [5] the complete Gro¨bner fan is considered, i.e., the whole Rn, sincethe considered toric ideals are homogeneous w.r.t. a certain grading. This is not thecase for our code ideal and so here the Gro¨bner fan is considered only in Rn+.Proposition 1. [4] Let  be a term order and v ∈C(I). For any u ∈ Rn holdsltu(I) = ltv(I) ⇐⇒ ltu(g) = ltv(g) ∀g ∈ G(I),where ltu stands for the leading (initial) term (ideal) induced by the order  givenby the weight vector u.Note that it is a well known fact that a reduced Gro¨bner basis for an ideal Iw.r.t. a certain monomial order is degree compatible if and only if the correspondingGro¨bner cone contains the all-one vector 1. From a coding-theory point of view,degree compatible orderings are the ones one must analyze since the weight of avector is translated on the degree of a monomial. In this sense degree compatibleorderings provide us a test set for the code and therefore a gradient descent decodingalgorithm, see [1]. The following proposition characterizes when there is a uniquedegree compatible Gro¨bner basis.Proposition 2. Let G be a reduced Gro¨bner basis for I(C ) w.r.t. a certain degreecompatible ordering. The Gro¨bner basis G is the only reduced degree compatibleGro¨bner basis for I(C ) if and only ifdeg(xa)> deg(xb) for all xa−xb ∈ G . (1)Proof. Assume that (1) holds but there is another Gro¨bner basis G ′ for I(C ) w.r.t.another degree compatible order′. Since′ is degree compatible, lt′(g) = lt(g)for all g ∈ G . And by Proposition 1 we see that lt′(I(C )) = lt(I(C )) and thus,G = G ′.Or equivalently, we can argue that the all-one vector is in the interior of the coneC(I(C )) and so clearly it cannot be contained in another cone in the Gro¨bner fan.In order to show the other direction assume that (1) does not hold, i.e., thereis at least one binomial xa − xb in G such that deg(xa) = deg(xb). Then 1 /∈Int(C(I(C ))) and in particular, there must be a neighbouring cone that also con-tains 1 and thus corresponds to a degree compatible ordering. uunionsqIn terms of the Gro¨bner fan the above proposition can also be expressed as fol-lows: A reduced Gro¨bner basis G w.r.t. a degree compatible ordering is the onlydegree compatible Gro¨bner basis if and only if the all-one vector 1 lies in the inte-rior of the Gro¨bner cone of G .4 Natalia Du¨ck, Irene Ma´rquez-Corbella and Edgar Martı´nez-MoroWe say that two binary linear codes C1 and C2 are permutation equivalent pro-vided there is a permutation of coordinates which sends C1 to C2. In the same fash-ion two binomial degree compatible Gro¨bner bases are permutation equivalent ifthere is a permutation of the variables that transforms one into the other. There isa close relationship between code equivalence and the equivalence of the degreecompatible Gro¨bner bases associated to their code ideals stated as follows: If thetwo degree compatible Gro¨bner bases are permutation equivalent so are the codes,unfortunately the converse is not true, given two permutation equivalent codes notall the degree compatible Gro¨bner bases are permutation equivalent (only two ofthem should be). The reader can see [2] for a proof of this discussion.Indeed if one has only a unique degree compatible Gro¨bner basis for a given code(Proposition 2) checking permutation equivalence is reduced to checking if the twounique bases are permutation equivalent using the techniques in [2]. If this is not thecase one needs to compute the whole set of degree compatible Gro¨bner bases whichwe call the degree compatible Gro¨bner fan. We will tackle this task in the followingsection.3 Adapting the TiGERS strategyWe can adapt the TiGERS Algorithm in [5] for computing the degree compatibleGro¨bner fan for I(C ) as follows: We start with a degree compatible Gro¨bner basis(note that this basis can be computed by the algorithm stated in [2]). By Proposi-tion 2 we can determine whether it is the only degree compatible Gro¨bner basis ornot. If not, we flip only those facet binomials where both terms have the same degreeand recompute the Gro¨bner basis. Unfortunately due the lack of space these stepscan not be detailed in this extended abstract but they are showed in [5]. Lemma 1below guarantees that we will always find at least one facet binomial where bothterms have the same degree. Additionally, we can employ the reverse search treedefined in [5] for traversing the Gro¨bner cones that are degree compatible.Lemma 1. Let G be the reduced Gro¨bner basis for I(C ) w.r.t. a degree com-patible ordering. If G is not the only degree compatible Gro¨bner basis, that is1 /∈ Int(C(I(C ))), then among all the facet binomials of G is at least one binomialxα −xβ such that deg(xα) = deg(xβ ).Proof. Let G ={xαi −xβi | 1≤ i≤ j+ k} and order the binomials such that deg(xαi)>deg(xβi) for 1≤ i≤ j and deg(xαi) = deg(xβi) for 1+ j ≤ i≤ j+ k.Assume that all facet binomials are such that the degree of the leading term isstrictly greater than the degree of the other term. Then the coneC′ ={u ∈ Rn+ | αi ·u≥ βi ·u for all 1≤ i≤ j}equals the Gro¨bner cone C(I(C )) of the Gro¨bner basis G . But then 1 ∈ Int(C′) =Int(C(I(C ))), which is a contradiction. uunionsqOn the fan associated to a linear code 5Lemma 2. Let Gnew be the reduced Gro¨bner basis obtained from Gold by flippingthe facet binomial xα − xβ . Any new leading terms in Gnew, i.e., leading terms ofGnew that do not appear in Gold , are divisible by xα .Proof. Any new leading terms arise from the Gro¨bner basis computation of thequasi-monomial idealT :={xβ −xα}∪T ′, T ′ :={xαi | xαi −xβi ∈ Gold}that consists of the designated flipping binomial with changed leading term and allthe other leading terms in Gold . To be more precise, a new leading term arises froman S-polynomial of the formS(xβ −xα ,xαi)= xγxα , where xγ = lcm(xβ ,xαi)/xβ ,which is not being reduced to zero by the elements in T . When computing a Gro¨bnerbasis, then this S-polynomial is either reduced to zero or its remainder on division bythe set T is added to the Gro¨bner basis of T . We distinguish the following situations:1. Neither xβ nor any monomial in T ′ divides xγxα : The monomial xγxα cannot befurther reduced and thus is being attached to the Gro¨bner basis of T .2. A monomial in T ′ divides xγxα : The monomial xγxα is being reduced to zeroand thus, this S-polynomial results in no new term.3. The monomial xβ divides xγxα : Since xα and xβ have disjoint support (see [1]),xβ has to divide xγ , the monomial xγxα is reduced toxγxα −xα xγxβ(xβ −xα)=xγxβ(xα)2 .So, whenever the S-polynomial cannot be reduced to zero, we obtain a monomialwhich is divisible by xα . uunionsqProposition 3. T(I(C )) is an acyclic directed graph with a unique sink that wewill call the reverse search tree.Proof. We prove that T(I(C )) is a tree by showing that there is no cycle in thisconstruction. We show this by contradiction. For the other claims see the proof of [5,Theorem 2.6].Assume that there is a cycle in the reverse search tree, say G1 −→ G2 −→ . . .−→G` −→ G1, where Gi+1 is obtained from Gi by flipping along xαi−xβi . Then Gi con-tains this binomial with leading term xαi and Gi+1 with leading term xβi . Inspectingthe cycle we see that the binomial xα1−xβ1 lies in G1 with leading term xα1 and ap-pears in G2 with leading term xβ1 . Then no binomial in G2 has the leading term xα1 .However, as we arrive at G1 after ` flipping steps, we conclude that xα1 −xβ1 mustbe inserted at some successive flipping step. Assume that this happens in the i1thflipping process, 1< i1 ≤ `. Then by Lem. 2, xα1 is divisible by xαi1 . And since G1is a Gro¨bner basis this implies that xαi1 cannot be the leading term of any element in6 Natalia Du¨ck, Irene Ma´rquez-Corbella and Edgar Martı´nez-MoroG1; it must have been inserted as a new leading term during some preceding flippingstep, say i2 < i1. By the same argument the monomial xαi1 is divisible by xαi2 andthen xαi2 cannot appear as the leading term of any element in G1. Continuing thisprocess we get a decreasing sequence of indices i1 > i2 > i3 > .. . which eventuallymust terminate, say after k steps, i.e., ik = 1. Then xαik = xα1 and from the divisi-bility relations xαik | xαik−1 | . . . | xαi2 | xαi1 | xα1 we actually obtain equality of allleading terms of the flipping binomials. However, this is a contradiction. uunionsqThe following proposition states the discussion at the end of Section 2.Proposition 4. Two linear codes C1 and C2 are permutation-equivalent if and onlyif they have the same degree compatible Gro¨bner fan structure, i.e., there is per-mutation σ ∈ Sn such that σ (G f an(C1)) =G f an(C2), where σ (G f an(C1)) meanspermuting the variables in each of the degree compatible Gro¨bner basis within thefan.Example 1. Consider two binary [6,3] codes C1 and C2 with respective parity checkmatricesH1 =1 1 0 0 0 00 0 1 1 0 00 0 0 0 1 1 , H2 = 1 1 1 1 1 10 0 0 1 0 10 1 0 1 0 0In [2, Example 2 and 5] it is shown that these codes are not permutation-equivalent.Here, we show how the degree compatible Gro¨bner fans of both codes can be em-ployed to show their non-equivalence. The degree compatible Gro¨bner fan for C1consists of 8 Gro¨bner basis which are all of cardinality 6 (see Ex. 2). The Gro¨bnerbasis for C2 w.r.t. the grevlex basis is given by{x3− x5,x1− x5,x4x5− x2x6,x2x5− x4x6,x2x4− x5x6} ∪{x2i −1 | i = 2,4,5,6}and consists of 9 elements. Thus, we can already conclude that these two codescannot be permutation-equivalent.Example 2. The reverse search tree T(I(C )) for the binary [6,3] code C1 from theprevious example with  being pure lex is given in Fig. 1.And the Gro¨bner bases are (the flipping binomials are underlined)G1 ={x1− x2,x3− x4,x5− x6,x22−1,x24−1,x26−1}G2 ={x1− x2,x4− x3,x5− x6,x22−1,x23−1,x26−1}G3 ={x2− x1,x4− x3,x5− x6,x21−1,x23−1,x26−1}G4 ={x1− x2,x3− x4,x6− x5,x22−1,x24−1,x25−1}G5 ={x1− x2,x4− x3,x6− x5,x22−1,x23−1,x25−1}G6 ={x2− x1,x4− x3,x6− x5,x21−1,x23−1,x25−1}G7 ={x2− x1,x3− x4,x6− x5,x21−1,x24−1,x25−1}G8 ={x2− x1,x3− x4,x5− x6,x21−1,x24−1,x26−1}.On the fan associated to a linear code 7G1G2 G4 G8G3 G5 G7G6Fig. 1 The reverse search tree for C1ConclusionsWe have shown how the computation of the degree compatible Gro¨bner fan of a codeis useful for determining the code equivalence problem. Anyway one can not forgetthat this is an NP-problem and therefore the Gro¨bner basis computation comprises ahard step. Further research in the topic points toward analyzing heuristic techniquesfor eliminating the need of transverse the whole fan or at least for trying to deducethe answer from partial information about the initial Gro¨bner basis.References1. Borges-Quintana, M., Borges-Trenard, M.A., Fitzpatrick, P., Martı´nez-Moro, E.: Gro¨bner basesand combinatorics for binary codes. Appl. Algebra Engrg. Comm. Comput. 19(5), 393–411(2008)2. Borges-Quintana, M., Borges-Trenard, M.A., Martı´nez-Moro, E.: On a Gro¨bner bases structureassociated to linear codes. J. Discrete Math. Sci. Cryptogr. 10(2), 151–191 (2007)3. Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms: An introduction to compu-tational algebraic geometry and commutative algebra. Undergraduate Texts in Mathematics.Springer-Verlag, New York, second edn. (1997)4. Fukuda, K., Jensen, A.N., Thomas, R.R.: Computing Gro¨bner fans. Math. Comp. 76(260),2189–2212 (electronic) (2007)5. Huber, B., Thomas, R.R.: Computing Gro¨bner fans of toric ideals. Experiment. Math. 9(3),321–331 (2000)6. Ma´rquez-Corbella, I., Martı´nez-Moro, E.: Algebraic structure of the minimal support code-words set of some linear codes. Adv. Math. Commun. 5(2), 233–244 (2011)7. Ma´rquez-Corbella, I., Martı´nez-Moro, E., Sua´rez Canedo, E.: On the ideal associated to anylinear code. Adv. Math. Commun. -(-), Submitted (2014)

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