Convolutional encoding of 60,64,68,72-bit self-dual codes

TL;DR

This paper presents a convolutional encoding method for specific self-dual codes with lengths 60, 64, 68, and 72, achieving minimal constraint length and introducing new code constructions.

Contribution

It introduces a novel convolutional encoding approach for these self-dual codes with minimal constraint length and presents new code parameters and constructions.

Findings

Successfully constructed self-dual codes with minimal constraint length K=9

Developed a pure quasi-cyclic construction with K=10 for certain codes

Discovered a new [72,36,12] singly even self-dual code with specific parameters

Abstract

In this paper we obtain the [60,30,12], [64,32,12], [68,34,12], [72,36,12] self-dual codes as tailbitting convolutional codes with the smallest constraint length K=9. In this construction one information bit is modulo two added to the one of the encoder outputs and the first row in the quasi-cyclic generator matrix is replaced by the obtained row. The pure quasi-cyclic construction with K=10 is also available for [68,34,12] and [72,36,12] codes. The new [72,36,12] singly even self-dual code with parameters Beta=483 Gamma=0 was obtained.