Norm-One Torus Decompositions and Decoding of Gashkov-Sidel'nikov Codes

2026-09-17

AI Summary

该研究解决了Gashkov-Sidel'nikov码的解码问题，通过将解码过程分为两个阶段，并利用二次特征和Weil界构造了完整的最大似然解码器。

Abstract

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.

Problem

• Gashkov-Sidel'nikov Codes
• Decoding
• Minimum Error Weight
• Syndrome
• Coset-Weight Distribution

Innovation

• norm-one torus
• Gashkov-Sidel'nikov codes
• quadratic character sums
• Weil bounds
• maximum-likelihood decoder