Self-Dual Codes, Symmetric Matrices, and Eigenvectors

Abstract

We introduce a consistent and efficient method to construct self-dual codes over GF (q) using symmetric matrices and eigenvectors from a self-dual code over GF (q) of smaller length where q equivalent to 1 (mod 4). Using this method, which is called a 'symmetric building-up' construction, we improve the bounds of the best-known minimum weights of self-dual codes with lengths up to 40, which have not significantly improved for almost two decades. We focus on a class of self-dual codes, which includes double circulant codes. We obtain 2967 new self-dual codes over GF (13) and GF (17) up to equivalence. We also compute the minimum weights of quadratic residue(QR) codes that were previously unknown. These are a [20,10,10] QR self-dual code over GF (23), [24,12,12] QR self-dual codes over GF (29) and GF (41), and a [32,16,14] QR self-dual code over GF (19). They have the highest minimum weights so far.