交换环上的矩阵乘积码及由$(σ,δ)$ -码产生的结构
Matrix-Product Codes over Commutative Rings and Constructions Arising from $(σ,δ)$-Codes
摘要：
证明了矩阵积码（MPC）的汉明距离的一个著名下界（在有限域和某些特殊的有限交换环上）在任何交换环$R$上都是有效的.一个充分条件，以及，这样的界限是尖锐的。当MPC的输入码都是自由码时，它是自由的，在这种情况下，给出了生成矩阵。如果$R$是有限的，一个充分条件提供了一个MPC的对偶是MPC，这种对偶的生成矩阵，并提出了LCD，自对偶，自正交MPC的特征。最后，本文的结果与作者以前的结果一起被用来构造新的MPC所产生的$(σ, δ)$ -代码。研究了这类结构的一些性质。
英文摘要：
A well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ring $R$. A sufficient condition is given, as well, for such a bound to be sharp. It is also shown that an MPC is free when its input codes are all free, in which case a generating matrix is given. If $R$ is finite, a sufficient condition is provided for the dual of an MPC to be an MPC, a generating matrix for such a dual is given, and characterizations of LCD, self-dual, and self-orthogonal MPCs are presented. Finally, results of this paper are used along with previous results of the authors to construct novel MPCs arising from $(σ, δ)$-codes. Some properties of such constructions are also studied.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
