环上码的广义权与单项理想的不变量
Generalized weights of codes over rings and invariants of monomial ideals
摘要：
本文发展了$R$ -定长线性码的支撑的代数理论，其中$R$是有限交换酉环。一个支持自然会导致广义权重的概念，并允许一个关联的单项式理想的代码。我们的主要结果表明，在适当的假设下，码的广义权可以从它的相关单项理想的分次Betti数得到。在$\mathbb{F}_q$ -具有汉明度量的线性码的情况下，理想与通过其奇偶校验矩阵与码相关联的拟阵的Stanley-Reisner理想一致。在这种特殊的设置，我们恢复已知的结果，广义的重量$\mathbb{F}_q$ -线性码可以得到的分次Betti数的理想的拟阵相关联的代码。我们还研究了码中的最小支集的子码和码字，证明了一大类$R$ -线性码是由其最小支集的码字生成的。
英文摘要：
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.
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