Optimum Subcodes of Self-Dual Codes and Their Optimum Distance Profiles

TL;DR

This paper develops algorithms to compute the optimum distance profiles of self-dual codes, determining these profiles for various code lengths and revealing new code structures and limitations.

Contribution

It introduces Chain Algorithms for finding ODPs and applies them to classify and analyze self-dual codes up to length 48, including new code examples and non-existence results.

Findings

ODPs computed for Type II codes up to length 24

ODPs determined for extremal Type II codes of length 32

First example of a doubly-even self-complementary [48,16,16] code

Abstract

Binary optimal codes often contain optimal or near-optimal subcodes. In this paper we show that this is true for the family of self-dual codes. One approach is to compute the optimum distance profiles (ODPs) of linear codes, which was introduced by Luo, et. al. (2010). One of our main results is the development of general algorithms, called the Chain Algorithms, for finding ODPs of linear codes. Then we determine the ODPs for the Type II codes of lengths up to 24 and the extremal Type II codes of length 32, give a partial result of the ODP of the extended quadratic residue code $q_{48}$ of length 48. We also show that there does not exist a $[48,k,16]$ subcode of $q_{48}$ for $k \ge 17$, and we find a first example of a doubly-even self-complementary $[48, 16, 16]$ code.