发表机构
Anhui University; University of Bergen; Sabancı University（安徽大学; 卑尔根大学; 萨班奇大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过范数一环面分解，将Gashkov-Sidel'nikov码的译码问题转化为最小加法长度计算，并利用二次特征和构造陪集首，实现最大似然译码。
AI 中文摘要
设$q=3^m$，令$K=\mathbb F_{q^2}$，并设\\[\mathcal T=\{x\in K^*:\operatorname{N}_{K/\mathbb F_q}(x)=1\}.\\]对于循环和常循环Gashkov-Sidel'nikov码，我们证明带符号校验列标签的集合恰好是$\mathcal T$。因此，译码问题分为两个阶段：确定与综合征$S$相关的最小错误权重，以及构造达到该最小值的错误向量。我们将前者与$S$关于$\mathcal T$的最小加法长度等同，并通过$\mathbb F_q$的范数和二次特征精确确定它。我们还确定了完整的陪集权重分布，并恢复了已知的覆盖半径$3$。对于构造部分，我们使用二次特征和与Weil界来为每个陪集权重为三的综合征构造陪集首。所得过程给出了完整的最大似然译码器。
英文摘要
Let $q=3^m$, let $K=\mathbb F_{q^2}$, and let \[\mathcal T=\{x\in K^*:\operatorname{N}_{K/\mathbb F_q}(x)=1\}.\] For both cyclic and constacyclic Gashkov-Sidel'nikov codes, we show that the set of signed parity-check column labels is precisely $\mathcal T$. Consequently, the decoding problem separates into two stages: determining the minimum error weight associated with a syndrome $S$ and constructing an error vector attaining this minimum. We identify the former quantity with the minimum additive length of $S$ with respect to $\mathcal T$ and determine it exactly by the norm and the quadratic character of $\mathbb F_q$. We also determine the complete coset-weight distribution and recover the known covering radius $3$. For the constructive part, we use quadratic-character sums and Weil bounds to construct a coset leader for every syndrome of coset weight three. The resulting procedures give complete maximum-likelihood decoders.
