论文标题
通过功能参数及其后果参数的几个几个重量线性代码的子场代码
Subfield Codes of Several Few-Weight Linear Codes Parametrized by Functions and Their Consequences
论文作者
论文摘要
有限字段上线性代码的子场代码最近受到了很多关注。其中一些代码是最佳的，并且在分泌共享，身份验证代码和关联方案中具有应用。在本文中，$ q $ -ary子字段代码$ c_ {f，g}^{（q）} $的六个不同系数系列$ c_ {f，g} $分别考虑了两个函数$ f，g $在有限字段上$ f_ {q^m} $参数化。 $ c_ {f，g}^{（q）} $的参数和（锤）的权重分布及其穿刺代码$ \ bar {c} _ {f，g}^{（q）} $被明确确定。还分析了这些代码双重的参数。结果$ q $ -ary代码$ c_ {f，g}^{（q）}，$ $ \ bar {c} _ {f，g}^{（q）} $，它们的双代码是最佳的，有些代码具有最著名的参数。线性代码的前两个系列$ c_ {f，g} $的参数和权重枚举者也被解决，其中第一个家族是符合Griesmer边界的最佳两次重量线性代码，这两个系列的双重代码几乎是MDS代码。作为本文的副产品，使用$ M \ geq 2 $获得了$ [2^{4M-2}的家族，2m+1,2^{4M-3}] $ quaternary hermitian自助式代码。作为一个应用程序，我们表明，派生线性代码的三个家族产生了几个无限的家族$ t $ -Designs（$ t \ in \ {2，3 \} $）。
Subfield codes of linear codes over finite fields have recently received much attention. Some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the $q$-ary subfield codes $C_{f,g}^{(q)}$ of six different families of linear codes $C_{f,g}$ parametrized by two functions $f, g$ over a finite field $F_{q^m}$ are considered and studied, respectively. The parameters and (Hamming) weight distribution of $C_{f,g}^{(q)}$ and their punctured codes $\bar{C}_{f,g}^{(q)}$ are explicitly determined. The parameters of the duals of these codes are also analyzed. Some of the resultant $q$-ary codes $C_{f,g}^{(q)},$ $\bar{C}_{f,g}^{(q)}$ and their dual codes are optimal and some have the best known parameters. The parameters and weight enumerators of the first two families of linear codes $C_{f,g}$ are also settled, among which the first family is an optimal two-weight linear code meeting the Griesmer bound, and the dual codes of these two families are almost MDS codes. As a byproduct of this paper, a family of $[2^{4m-2},2m+1,2^{4m-3}]$ quaternary Hermitian self-dual code are obtained with $m \geq 2$. As an application, we show that three families of the derived linear codes give rise to several infinite families of $t$-designs ($t \in \{2, 3\}$).
