Research on the Construction of Maximum Distance Separable Codes via Arbitrary twisted Generalized Reed-Solomon Codes
Chun'e Zhao, Wenping Ma, Tongjiang Yan, Yuhua Sun
TL;DR
The paper addresses when arbitrary-twist generalized Reed-Solomon (A-TGRS) codes are MDS by deriving an explicit inverse for the Vandermonde matrix $V$ and an equivalent MDS condition. This framework shows that all known MDS-TGRS constructions are special cases of A-TGRS and enables the discovery of new MDS-TGRS families with novel parameter matrices $A(oldsymbol{ heta})$. Additionally, the work connects the inverse of lower triangular Toplitz matrices to a linear feedback shift register (LFSR) process, providing a practical computational tool. The results yield both theoretical unification of prior MDS-TGRS criteria and practical methods for parity-check simplification and code construction over finite fields, with broad relevance to cryptography and error correction.
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.