Self-Dual Codes, Symmetric Matrices, and Eigenvectors

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.