利用自由分辨率计算二进制线性码的广义汉明权重

IF 0.4 Q4 MATHEMATICS, APPLIED ACM Communications in Computer Algebra Pub Date : 2022-06-01 DOI:10.1145/3572867.3572868
Ignacio García-Marco, Irene Márquez-Corbella, E. Martínez-Moro, Yuriko Pitones
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引用次数: 0

摘要

在这项工作中,我们探索了一些单项式理想的自由分辨率与二进制码的广义汉明权重(GHW)之间的关系。更准确地说,我们寻找一个比最小支持的码字集更小的结构,为我们提供一些关于GHW的信息。我们证明了二元线性码的第一和第二广义汉明权可以(通过分级自由分辨率)从与该码相关的二项式理想相关的一组单项式中计算出来。此外,剩余的权重受该集合的Betti数的限制。
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Computing generalized hamming weights of binary linear codes via free resolutions
In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated to a binomial ideal related with the code. Moreover, the remaining weights are bounded by the Betti numbers for that set.
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