Gröbner bases and the second generalized Hamming weight of a linear code
Hernán de Alba (1 and 2), Cecilia Martínez-Reyes (2) ((1) SECIHTI, (2) Universidad Autónoma de Zacatecas)
众所周知,对于二进制代码,可以使用Gröbner碱基获得最小支持的码数子集,可用于确定代码的第二个广义Hamming权重。 在本文中,我们根据非二进制代码建立了条件,其中同一属性持有。 我们还在任何属性不持有的非二进制有限字段上构建一个代码家族。 此外,我们证明,每当通过Gröbner基础获得的子集足以确定第二个广义Hamming重量时,这种不变性也可以从最小自由分辨率的syzygies度中恢复。
It is known that for binary codes one can use Gröbner bases to obtain a subset of codewords of minimal support that can be used to determine the second generalized Hamming weight of the code. In this paper we establish conditions on a nonbinary code under which the same property holds. We also construct a family of codes over any nonbinary finite field where the property does not hold. Furthermore, we prove that whenever the subset obtained via Gröbner basis suffices to determine the second gene...
