Quasi-cyclic self-dual codes of length 70

TL;DR

This paper constructs new quasi-cyclic self-dual codes of length 70, including the first known code with specific parameters, expanding the understanding of self-dual code structures.

Contribution

It introduces new quasi-cyclic self-dual codes of length 70, notably the first with parameters Beta=140 Gamma=0, and shows they are not pure double circulant.

Findings

Constructed multiple [70,35,12] codes as quasi-cyclic codes

Identified a unique code with Beta=140, Gamma=0

Demonstrated these codes are not pure double circulant

Abstract

In this paper we obtain a number of [70,35,12] singly even self-dual codes as a quasi-cyclic codes with m=2 (tailbitting convolutional codes). One of them is the first known code with parameters Beta=140 Gamma=0. All codes are not pure double circulant i.e. could not be represented in systematic form.