On the Hamming Distance and LCD Properties of Binary Polycyclic Codes and Their Duals

TL;DR

This paper investigates the algebraic structure, minimum Hamming distance, duality, and LCD properties of binary polycyclic codes, providing exact distances, bounds, and conditions for LCD status, especially for codes related to self-reciprocal irreducible trinomials.

Contribution

It offers a comprehensive analysis of binary polycyclic codes, including new results on their structure, distance, duals, and LCD properties, with explicit constructions and a conjecture on their LCD nature.

Findings

Determined the exact Hamming distance for codes associated with specific irreducible trinomials.

Established necessary and sufficient conditions for binary polycyclic codes to be LCD.

Constructed families of optimal and LCD optimal binary linear codes, including larger lengths.

Abstract

Polycyclic codes offer a natural generalization of cyclic codes and provide a broader algebraic framework for constructing linear codes with good parameters. In this paper, we study binary polycyclic codes associated with powers of irreducible polynomials. We first determine their complete algebraic structure and then develop general results on their minimum Hamming distance, including several exact values and bounds. We also examine the Euclidean duals of these codes and derive corresponding results on the Hamming distance of the dual codes. Furthermore, we study the LCD (linear complementary dual) properties of binary polycyclic codes, establish necessary and sufficient conditions for such codes to be LCD codes, and construct several families of binary LCD codes. Our constructions also yield many optimal and LCD optimal binary linear codes, including codes of larger lengths. We then focus on binary polycyclic codes associated with powers of the self-reciprocal irreducible trinomials $x^{2\cdot3^v}+x^{3^v}+1$, where $v\geq0$. For this class, we determine the exact Hamming distance of all such codes and show that these codes are reversible. Moreover, we show that these codes are LCD codes in certain cases. In addition, we propose a conjecture asserting that all binary polycyclic codes associated with $\big(x^{2\cdot3^v}+x^{3^v}+1\big)^{2^\mathcal{T}}$, where $v\geq 0$ and $\mathcal{T}\geq1$, are LCD codes. These results demonstrate that binary polycyclic codes form a rich source of structured codes with strong distance, duality, reversibility, and LCD properties.