Twisted reed solomon codes have fully mapped weight hierarchies
Generalized Hamming Weights and Subcode Support Weight Distributions of Twisted Reed-Solomon Codes
Information Theory
Summary
This paper studies specialized error-correcting codes called twisted Reed-Solomon codes. The authors figured out all possible ways these codes can measure error detection strength, known as generalized Hamming weights. They provide formulas that help compute these measurements for codes with one or two twists, which helps understand the code’s reliability and structure. For a particular class, they even describe how many smaller code segments have specific error properties.
What this means in practice
- •For communications engineers: Design error-correcting codes with predictable error detection capabilities for more reliable data transmission.
- •For data storage system developers: Select twisted Reed-Solomon codes with known subcode structures to optimize fault tolerance in storage devices.
Authors
Chao Liu, Hao Chen, Dabin Zheng, Yaqi Chen
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.