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

2026-05-12

AI Summary

This study investigates the Hamming distance and linear complementary dual (LCD) properties of binary polycyclic codes generated by powers of irreducible polynomials and their duals. By leveraging algebraic coding theory, structural analysis of polynomial rings, and dual code techniques, the work provides the first complete characterization of the algebraic structure of such codes, precisely determines their minimum Hamming distance, and establishes tight bounds. It also derives necessary and sufficient conditions for these codes to be LCD. Key contributions include the construction of several optimal and LCD-optimal binary linear codes, a proof of code reversibility under specific conditions, and a conjecture that all codes generated by powers of self-reciprocal irreducible trinomials are LCD, thereby expanding the known sources for constructing optimal LCD codes.

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.

Problem

• polycyclic codes
• Hamming distance
• LCD codes
• dual codes
• binary linear codes

Innovation

• polycyclic codes
• Hamming distance
• LCD codes
• dual codes
• self-reciprocal polynomials