论文标题
戒指和单一理想不变的代码的广义权重
Generalized weights of codes over rings and invariants of monomial ideals
论文作者
论文摘要
我们开发了固定长度的$ r $ - 线性代码的代数支持理论，其中$ r $是有限的交换统一戒指。支持自然会引起广义权重的概念，并允许人们将单一理想与代码联系起来。我们的主要结果指出，在适当的假设下，可以从其相关单式理想的分级betti数字中获得代码的广义权重。如果是$ \ mathbb {f} _q $ - 带有锤式指标的线性代码，理想的理想与Stanley-Reisner通过其奇偶校验检查矩阵相关的Matroid的理想是相吻合的。在这种特殊环境中，我们恢复了已知的结果，即可以从与代码相关的Matroid的理想的分级betti数字中获得$ \ mathbb {f} _q $ - 线性代码的广义权重。我们还研究了代码中最小支持的子代码和代码字，证明了大量的$ r $ - 线性代码是由其最小支持的代码人产生的。
We develop an algebraic theory of supports for $R$-linear codes of fixed length, where $R$ is a finite commutative unitary ring. A support naturally induces a notion of generalized weights and allows one to associate a monomial ideal to a code. Our main result states that, under suitable assumptions, the generalized weights of a code can be obtained from the graded Betti numbers of its associated monomial ideal. In the case of $\mathbb{F}_q$-linear codes endowed with the Hamming metric, the ideal coincides with the Stanley-Reisner ideal of the matroid associated to the code via its parity-check matrix. In this special setting, we recover the known result that the generalized weights of an $\mathbb{F}_q$-linear code can be obtained from the graded Betti numbers of the ideal of the matroid associated to the code. We also study subcodes and codewords of minimal support in a code, proving that a large class of $R$-linear codes is generated by its codewords of minimal support.
