Decoding Algorithms for Symbol-Error Correction in MDS Array Codes via Superregular Matrices

Abstract

Maximum distance separable (MDS) array codes constitute an important class of error-correcting codes due to their optimal distance properties and their relevance in distributed storage systems. In this paper, we investigate the construction and decoding of MDS array codes over $\mathbb{F}_q^b$ based on superregular matrices, with emphasis on superregular Vandermonde and Cauchy matrices. We propose decoding algorithms for [n,k,d] MDS array codes, where n=m+k and d=m+1, capable of correcting symbol errors without prior knowledge of their locations. Unlike existing approaches restricted to specific parameter settings, the proposed algorithms apply to general configurations and rely on algebraic relations that do not follow from straightforward extensions of previous methods. Specifically, these algorithms correct one symbol error for $m \geq 2$ and two symbol errors for $m \geq 4$. For the two-error case, the decoding procedure admits a simplified form when Vandermonde superregular matrices are employed, reducing computational complexity. We analyze the algebraic structure of the three-symbol-error case, focusing on the most involved configuration in which all errors occur in information symbols, and we discuss how the method may be extended to the general case. These algorithms are computationally efficient for moderate parameter sizes, as they rely on structured algebraic operations over $\mathbb{F}_q^b$ and the solution of small linear systems, making them suitable for distributed storage applications. From an application perspective, the proposed approach provides a flexible alternative to RAID~6 schemes. Unlike RAID~6, which is limited to two parity disks and often requires prior knowledge of error locations, our construction supports general configurations and enables the correction of multiple symbol errors without location information, at the cost of increased algebraic complexity