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

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.