New extremal binary self-dual codes of lengths 64 and 66 from bicubic planar graphs

TL;DR

This paper constructs new extremal binary self-dual codes of lengths 64 and 66 using bipartite graphs and lifting techniques, expanding the known catalog of such codes with specific weight enumerators.

Contribution

It introduces a novel method of deriving extremal self-dual codes from bipartite graphs and their lifts, resulting in 25 new codes of lengths 64 and 66.

Findings

15 new codes of length 64 with specific weight enumerators

10 new codes of length 66 with unique weight enumerators

First construction of codes with these weight enumerators in literature

Abstract

In this work, connected cubic planar bipartite graphs and related binary self-dual codes are studied. Binary self-dual codes of length 16 are obtained by face-vertex incidence matrices of these graphs. By considering their lifts to the ring R_2 new extremal binary self-dual codes of lengths 64 are constructed as Gray images. More precisely, we construct 15 new codes of length 64. Moreover, 10 new codes of length 66 were obtained by applying a building-up construction to the binary codes. Codes with these weight enumerators are constructed for the first time in the literature. The results are tabulated.