Characterizations and Constructions of Linear Intersection Pairs of Cyclic Codes over Finite Fields

TL;DR

This work characterizes and constructs linear $\ell$-intersection pairs of cyclic codes over finite fields by linking intersection structure to generator polynomials and the factorization of $x^n-1$ via cyclotomic cosets. It establishes existence criteria through the degree of the least common multiple of generators and provides constructive schemes using a monic divisor $L(x)$, along with gcd-based refinements; it also shows how to realize these pairs in the MDS regime using Reed-Solomon codes. Numerical examples over $\mathbb{F}_2$ illustrate optimal or near-optimal parameter pairs, validating the theory and its potential for cryptographic and quantum error-correction applications. The results offer a practical framework for designing cyclic-code pairs with prescribed intersections and good performance characteristics.

Abstract

Linear intersection pairs of linear codes have become of interest due to their nice algebraic properties and wide applications. In this paper, we focus on linear intersection pairs of cyclic codes over finite fields. Some properties of cyclotomic cosets in cyclic groups are presented as key tools in the study of such linear intersection pairs. Characterization and constructions of two cyclic codes of a fixed intersecting dimension are given in terms of their generator polynomials and cyclotomic cosets. In some cases, constructions of two cyclic codes of a fixed intersecting subcode are presented as well. Based on the theoretical characterization, some numerical examples of linear intersection pairs of cyclic codes with good parameters are illustrated.

Theorems & Definitions (24)

• Theorem 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 3.1
• Lemma 3.2
• Proposition 3.3
• proof
• Corollary 3.4
• Theorem 3.5
• proof