Extremal Self-Dual Codes and Linear Complementary Dual Codes from Double Circulant Codes

TL;DR

This paper investigates the properties of extremal self-dual and LCD double circulant codes over GF(2), establishing conditions for their self-duality, extremality, and LCD status, with a focus on codes up to length 44.

Contribution

It provides necessary and sufficient conditions for self-duality and extremality of double circulant codes, and introduces criteria for bordered DC codes to be LCD over GF(2).

Findings

Conditions for self-duality of DC codes up to length 44.

Characterization of extremal and non-extremal self-dual DC codes.

Sufficient conditions for bordered DC codes to be LCD.

Abstract

This paper explores extremal self-dual double circulant (DC) codes and linear complementary dual (LCD) codes of arbitrary length over the Galois field $\mathbb F_2$. We establish the sufficient and necessary conditions for DC codes and bordered DC codes to be self-dual and identify the conditions for self-dual DC codes of length up to 44 to be extremal or non-extremal. Additionally, The self-duality and extremality between DC codes and bordered DC codes are also examined. Finally, sufficient conditions for bordered DC codes to be LCD codes over $\mathbb F_2$ under Euclidean inner product are presented.