Generalized Hamming Weights and Subcode Support Weight Distributions of Twisted Reed-Solomon Codes

Abstract

In this paper, we determine the generalized Hamming weight hierarchies of two classes of twisted Reed--Solomon (TRS) codes. For TRS codes with one twist, we first determine all possible values of the generalized Hamming weights. We then give a complete characterization of the weight hierarchy in terms of the minimum distance of the dual code. Explicit conditions are derived for the minimum distances of the code and its dual, which make the whole weight hierarchy explicitly computable. As special cases, the obtained results recover several known characterizations for TRS codes to be MDS, NMDS codes, and $s$-gap AMDS codes. For TRS codes with two twists, we similarly determine the complete generalized Hamming weight hierarchy. In this case, the higher generalized Hamming weights are determined by the first two generalized Hamming weights of the dual code, for which explicit rank conditions are also obtained. In addition, we further determine the subcode support weight distributions of this class of single-twist TRS codes. More precisely, for each $1 \le r \le k$, we derive an explicit formula for the number of $r$-dimensional subcodes with a prescribed support size, thereby giving a complete description of the support distribution of subcodes.