An Efficient Construction of Self-Dual Codes

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 $[24,12,9]$.