Research on the Construction of Maximum Distance Separable Codes via Arbitrary twisted Generalized Reed-Solomon Codes

TL;DR

This paper investigates conditions under which twisted generalized Reed-Solomon (TGRS) codes are maximum distance separable (MDS), providing explicit formulas, new classes of MDS TGRS codes, and a novel method for matrix inversion.

Contribution

It introduces a general class of TGRS codes, derives explicit inverse formulas, and identifies new MDS TGRS codes with broader parameters than previously known.

Findings

Derived explicit inverse of Vandermonde matrix.

Established conditions for TGRS codes to be MDS.

Presented a new method for inverting Toeplitz matrices.

Abstract

Maximum distance separable (MDS) codes have significant combinatorial and cryptographic applications due to their certain optimality. Generalized Reed-Solomon (GRS) codes are the most prominent MDS codes. Twisted generalized Reed-Solomon (TGRS) codes may not necessarily be MDS. It is meaningful to study the conditions under which TGRS codes are MDS. In this paper, we study a general class of TGRS (A-TGRS) codes which include all the known special ones. First, we obtain a new explicit expression of the inverse of the Vandermonde matrix. Based on this, we further derive an equivalent condition under which an A-TGRS code is MDS. According to this, the A-TGRS MDS codes include nearly all the known related results in the previous literatures. More importantly, we also obtain many other classes of MDS TGRS codes with new parameter matrices. In addition, we present a new method to compute the inverse of the lower triangular Toplitz matrix by a linear feedback shift register, which will be very useful in many research fields.