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 $q \equiv 1 \pmod 4$ . 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.

Keywords

• Eigenvectors
• optimal codes
• quadratic residue codes
• self-dual codes
• symmetric matrix
• symmetric self-dual codes