# An Efficient Construction of Self-Dual Codes. (arXiv:1201.5689v1 [cs.IT])

from cs.IT updates on arXiv.org http://arxiv.org/abs/1201.5689

We complete the building-up construction for self-dual codes by resolving the
open cases over $GF(q)$ with $q \equiv 3 \pmod 4$, and over $\Z_{p^m}$ and
Galois rings $\GR(p^m,r)$ with an odd prime $p$ satisfying $p \equiv 3 \pmod 4$
with $r$ odd. We also extend the building-up construction for self-dual codes
to finite chain rings. Our building-up construction produces many new
interesting self-dual codes. In particular, we construct 945 new extremal
self-dual ternary $[32,16,9]$ codes, each of which has a trivial automorphism
group. We also obtain many new self-dual codes over $\mathbb Z_9$ of lengths
$12, 16, 20$ all with minimum Hamming weight 6, which is the best possible
minimum Hamming weight that free self-dual codes over $\Z_9$ of these lengths
can attain. From the constructed codes over $\mathbb Z_9$, we reconstruct
optimal Type I lattices of dimensions $12, 16, 20,$ and 24 using Construction
$A$; this shows that our building-up construction can make a good contribution
for finding optimal Type I lattices as well as self-dual codes. We also find
new optimal self-dual $[16,8,7]$ codes over GF(7) and new self-dual codes over
GF(7) with the best known parameters $[24,12,9]$.