An Efficient Construction of Self-Dual Codes

TL;DR

This paper advances the construction of self-dual codes over various rings and fields, producing numerous new codes with optimal or extremal properties, and contributes to the understanding of their applications in lattice theory.

Contribution

The paper completes the building-up construction for self-dual codes over specific rings and fields, and introduces extensions that generate many new extremal and optimal self-dual codes.

Findings

Constructed 945 new extremal self-dual ternary [32,16,9] codes

Generated many new self-dual codes over  with lengths 12, 16, 20, minimum weight 6

Reconstructed optimal Type I lattices from codes over 

Abstract

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 $[24,12,9]$.