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

TL;DR

The work develops algebraic criteria for extremal self-dual and LCD properties of double circulant and bordered double circulant codes over $\mathbb{F}_2$, expressing self-duality through the polynomial condition $f(x)\overline{f(x)}\equiv 1$ mod $x^m-1$ and exploiting equivalences under $f^*(x)$ and $x^i f(x)$. It provides explicit extremal constructions for lengths up to 20 and partial results up to 44, along with a comprehensive treatment of bordered DC codes, including a gcd-based LCD condition and parity constraints linking code parameters. The results enable systematic construction of extremal and LCD codes without exhaustive search and establish a framework extendable to longer lengths and other fields. The methods hinge on orthogonal circulant matrices, generator polynomials, and modular polynomial arithmetic, with computational verification via MAGMA. These contributions advance the design of extremal self-dual and LCD codes in the DC code family and offer practical criteria for code construction in binary settings.

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.

Theorems & Definitions (53)

• Lemma 3.1
• Lemma 3.2
• proof
• Remark 3.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• Lemma 3.4