Error correction by Reed–Solomon codes using its automorphisms

Abstract

The article explores the syndrome invariants of АГ-group of automorphisms of Reed–Solomon codes (RS-codes) that are a joint group of affine and cyclic permutations. The found real invariants are a set of norms of N Г-orbits that make up one or another АГ-orbit. The norms of Г-orbits are vectors with 2 1 Cδ− coordinates from the Galois field, that are determined by all kinds of pairs of components of the error syndromes. In this form, the invariants of the АГ-orbits were cumbersome and difficult to use. Therefore, their replacement by conditional partial invariants is proposed. These quasi-invariants are called norm-projections. Norm-projection uniquely identifies its АГ-orbit and therefore serves as an adequate way for formulating the error correction method by RS-codes based on АГ-orbits. The power of the АГ-orbits is estimated by the value of N2, equal to the square of the length of the RS-code. The search for error vectors in transmitted messages by a new method is reduced to parsing the АГ‑orbits, but actually their norm-projections, with the subsequent search for these errors within a particular АГ-orbit. Therefore, the proposed method works almost N2 times faster than traditional syndrome methods, operating on the basic of the “syndrome – error” principle, that boils down to parsing the entire set of error vectors until a specific vector is found.