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

Zohreh Aliabadi, Cem Güneri, Tekgül Kalaycı

TL;DR

This work analyzes the hull and complementarity properties of two quasi-cyclic code families: one-generator QC codes and four-circulant codes. It develops explicit hull-dimension formulas expressed in terms of generating and parity-check polynomials, and provides complete LCD and LCP characterizations by leveraging a CRT-based decomposition into constituents. Notably, FC codes have even hull dimensions, and DC codes cannot have hull dimension 1 when $q \equiv 3 \pmod{4}$; the paper also gives necessary conditions (and counterexamples) for LCD status and describes LCP criteria for both classes. Computational results for binary and ternary cases illustrate near-optimal LCD/LCP parameters and hull-1 hull behavior, highlighting practical implications for cryptography and quantum-code constructions.

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.

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

TL;DR

This work analyzes the hull and complementarity properties of two quasi-cyclic code families: one-generator QC codes and four-circulant codes. It develops explicit hull-dimension formulas expressed in terms of generating and parity-check polynomials, and provides complete LCD and LCP characterizations by leveraging a CRT-based decomposition into constituents. Notably, FC codes have even hull dimensions, and DC codes cannot have hull dimension 1 when ; the paper also gives necessary conditions (and counterexamples) for LCD status and describes LCP criteria for both classes. Computational results for binary and ternary cases illustrate near-optimal LCD/LCP parameters and hull-1 hull behavior, highlighting practical implications for cryptography and quantum-code constructions.

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 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.
Paper Structure (4 sections, 9 theorems, 51 equations, 8 tables)

This paper contains 4 sections, 9 theorems, 51 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. One-Generator QC Codes
  4. Four-Circulant Codes

Key Result

Theorem 3.1

Let $C=\langle (a_1(x), \ldots, a_\ell(x))\rangle$ be a 1-generator $\ell$-QC code of length $m\ell$ over $\mathbb{F}_q$, whose parity check polynomial is $h(x)$. Then the hull dimension of $C$ is $h(C)=\deg u(x)$, where

Theorems & Definitions (18)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • proof
  • ...and 8 more