PERSYMMETRIC MATRIX AND ITS APPLICATION IN CODING THEORY

Abstract

A persymmetric matrix is a square matrix that is symmetric concerning its antidiagonal. This article discusses some characteristics of a persymmetric matrix and its application in coding theory. A persymmetric matrix is used to form a generator matrix of binary reversible self-dual codes. A binary reversible self-dual code is a self-dual code whose reverse element is contained in the code. The methodology involves the implementation of flip transpose and column reversal to ensure the generator matrix satisfies both self-duality and reversibility. We begin with small-sized persymmetric matrices (e.g., 2×2 and 3×3) to extend the construction of the larger matrices. Combining a self-dual code and a reversible self-dual code of shorter length, and embedding persymmetric blocks, we construct new binary reversible self-dual codes of longer length. The novelty of this research lies in developing a new construction method for binary reversible self-dual codes derived directly from self-dual codes in the standard form, which has not been explicitly addressed in previous studies.