Column Twisted Reed-Solomon Codes as MDS Codes

TL;DR

This paper introduces column twisted Reed-Solomon codes as a new method for constructing MDS codes with more flexible parameters, differing from traditional RS and TGRS codes, and analyzes their properties.

Contribution

It establishes conditions for column TRS codes to be MDS, shows they are not equivalent to RS codes, and extends parameter ranges beyond existing TGRS constructions.

Findings

Column TRS codes can be MDS under certain conditions.

These codes are not equivalent to Reed-Solomon codes.

Support for longer code lengths up to (q+3)/2 for large q.

Abstract

In this paper, we study column twisted Reed-Solomon(TRS) codes. We establish some sufficient conditions for these codes to be MDS and show that the dimension of their Schur square codes is $2k$. Consequently, these TRS codes are shown to be not equivalent to Reed-Solomon(RS) codes. Moreover, our construction offers more flexible parameters than existing twisted generalized Reed-Solomon(TGRS) code designs. For a large odd prime power $q$, systematically constructed TGRS codes are known to be limited to length $\frac{q+1}{2}$. By contrast, our column TRS construction supports code lengths up to $\frac{q+3}{2}$. Finally, we present the dual codes of column TRS codes. Overall, this work introduces a new method for constructing MDS codes by appending column vectors to some generator matrix of an RS code.