On the hull and complementarity of one generator quasi-cyclic codes and four-circulant codes

TL;DR

This paper analyzes the hull dimensions and complementarity properties of one generator quasi-cyclic codes and four-circulant codes, providing theoretical results and computational insights into their structure and limitations.

Contribution

It characterizes hull dimensions and linear complementary pairs for these code classes, revealing new structural properties and restrictions based on algebraic parameters.

Findings

Hull dimension of four-circulant codes is even

One-dimensional hulls are impossible for double-circulant codes when q ≡ 3 mod 4

Computational results support theoretical characterizations

Abstract

We study one generator quasi-cyclic codes and four-circulant codes, which are also quasi-cyclic but have two generators. We state the hull dimensions for both classes of codes in terms of the polynomials in their generating elements. We prove results such as the hull dimension of a four-circulant code is even and one-dimensional hull for double-circulant codes, which are special one generator codes, is not possible when the alphabet size $q$ is congruent to 3 mod 4. We also characterize linear complementary pairs among both classes of codes. Computational results on the code families in consideration are provided as well.