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A method for constructing quaternary Hermitian self-dual codes and an application to quantum codes

Masaaki Harada

TL;DR

This work develops the modified four μ-circulant construction as a practical method to build quaternary Hermitian self-dual codes over $F_4$ with large minimum weights. By deriving a concrete Hermitian self-duality criterion and exploiting equivalences, the authors perform extensive computer-search classifications and produce new extremal or near-extremal codes up to length $n=80$, including the first $[56,28,16]$ examples. They also determine the weight enumerators for the new $[56,28,16]$ codes and push the known bounds on maximum minimum weight for several lengths, while illustrating direct applicability to quantum coding via the CRSS framework to yield a $[[56,0,16]]$ quantum code. These results advance both classical code construction and quantum-code design, providing explicit constructions and data for further optimization and analysis.

Abstract

We introduce quaternary modified four $μ$-circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four $μ$-circulant Hermitian self-dual codes. We also construct quaternary modified four $μ$-circulant Hermitian self-dual codes having large minimum weights. Two quaternary Hermitian self-dual $[56,28,16]$ codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) $[56,28]$ codes. In addition, these codes imply the existence of a quantum $[[56,0,16]]$ code.

A method for constructing quaternary Hermitian self-dual codes and an application to quantum codes

TL;DR

This work develops the modified four μ-circulant construction as a practical method to build quaternary Hermitian self-dual codes over with large minimum weights. By deriving a concrete Hermitian self-duality criterion and exploiting equivalences, the authors perform extensive computer-search classifications and produce new extremal or near-extremal codes up to length , including the first examples. They also determine the weight enumerators for the new codes and push the known bounds on maximum minimum weight for several lengths, while illustrating direct applicability to quantum coding via the CRSS framework to yield a quantum code. These results advance both classical code construction and quantum-code design, providing explicit constructions and data for further optimization and analysis.

Abstract

We introduce quaternary modified four -circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four -circulant Hermitian self-dual codes. We also construct quaternary modified four -circulant Hermitian self-dual codes having large minimum weights. Two quaternary Hermitian self-dual codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) codes. In addition, these codes imply the existence of a quantum code.
Paper Structure (11 sections, 10 theorems, 20 equations, 12 tables)

This paper contains 11 sections, 10 theorems, 20 equations, 12 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quaternary codes
  4. Quaternary Hermitian self-dual codes
  5. Definition and basic properties of modified four $\mu$-circulant codes
  6. Numerical results of modified four $\mu$-circulant Hermitian self-dual codes
  7. Classification of modified four $\mu$-circulant Hermitian self-dual codes
  8. Largest minimum weights of modified four $\mu$-circulant Hermitian self-dual codes
  9. $C_{56,1}$ and $C_{56,\omega}$
  10. Largest minimum weights $d(n)$
  11. Application to quantum codes

Key Result

Lemma 1

Let $C$ be a quaternary $[2n,n]$ code with generator matrix $\left( \right)$. If $M \overline{M}^T=I_n$, then $C$ is Hermitian self-dual.

Theorems & Definitions (16)

  • Lemma 1: MOSW
  • Lemma 2
  • proof
  • Remark 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 6 more