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Item type:Publication, Access status: with files , Graver bases and niversal Gröbner bases for linear codesA linear code over a finite field can be associated with two binomial ideals. In this paper, algorithms for computing their Graver bases and universal Gröbner bases are given.Publicationtype: Journal Article251 298 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Singleton codesEach linear code can be described by a so-called code ideal. In order to utilize this ideal, Gröbner bases are required. Since many results depend on the chosen term order, knowledge of the universal Gröbner basis is advantageous. Singleton codes have the property that the universal Gröbner basis for their code ideals consists of all binomials associated to a codeword whose Hamming weight satisfies the Singleton bound. In this paper, properties of Singleton codes will be established and it will be examined which classical binary linear codes belong to the class of Singleton codes.Publicationtype: Journal Article33 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Vector space bases for the homogeneous parts in homogeneous ideals and graded modules over a polynomial ring© 2014 Academic Publications, Ltd. In this paper, vector space bases for the homogeneous parts of homogeneous ideals and graded modules over a commutative polynomial ring are given using Gröbner bases.Publicationtype: Journal Article223 422 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Computing generating sets for quaternary codes using Gröbner basesGröbner bases techniques can be used to compute a basis of a subspace of a finite-dimensional vector space over finite prime field given as a matrix kernel. Linear codes can be described as such subspaces and thus are an interesting area of application. Based on this, Gröbner bases techniques will be used to compute a generating set of a quaternary code given as a matrix kernel. In particular, if the quaternary code is free, the algorithm will provide a basis for the dual code.Publicationtype: Journal Article237 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Universal Gröbner bases for binary linear codesEach linear code can be described by a code ideal given as the sum of a toric ideal and a non-prime ideal. In this way, several concepts from the theory of toric ideals can be translated into the setting of code ideals. It will be shown that after adjusting some of these concepts, the same inclusion relationship between the set of circuits, the universal Gröbner basis and the Graver basis holds. Furthermore, in the case of binary linear codes, the universal Gröbner basis will consist of all binomials which correspond to codewords that satisfy the Singleton bound and a particular rank condition. This will give rise to a new class of binary linear codes denoted as Singleton codes. © 2013 Academic Publications, Ltd.Publicationtype: Journal Article175 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Groebner bases for linear codes over GF(4)A linear code over a prime field can be described by a binomial ideal in a polynomial ring given as the sum of a toric ideal and a nonprime ideal. A Groebner basis for such an ideal can be read off from a systematic generator matrix of the corresponding code. In this paper, a similar result will be presented for linear codes over GF(4). To this end, the extented alphabet GF(4) is dealt with by enlarging the polynomial ring. © 2011 Academic Publications, Ltd.Publicationtype: Journal Article14 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Groebner bases for quaternary codesA linear code can be described by a binomial ideal in a polynomial ring, given as the sum of a toric ideal and a nonprime ideal. A Groebner basis for such an ideal can be read off from a systematic generator matrix for the corresponding code. In this paper, an analogue result will be presented for quaternary codes. © 2011 Academic Publications, Ltd.Publicationtype: Journal Article10