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arXiv:2401.15265 · Designs Codes and Cryptography

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

Masaaki Harada

Published May 21, 2024• 1 citation
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Abstract

Abstract We introduce quaternary modified four $$\mu $$ μ -circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four $$\mu $$ μ -circulant Hermitian self-dual codes. We also construct quaternary modified four $$\mu $$ 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.

DOI
10.1007/s10623-024-01421-x

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Abstract We introduce quaternary modified four μ\mu μ μ -circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four μ\mu μ μ -circulant Hermitian self-dual codes. We also construct quaternary modified four μ\mu μ 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.
We present new constructions of binary quantum codes from quaternary linear Hermitian self-dual codes. Our main ingredients for these constructions are nearly self-orthogonal cyclic or duadic codes over F_4. An infinite family of 000-dimensional binary quantum codes is provided. We give minimum distance lower bounds for our quantum codes in terms of the minimum distance of their ingredient linear codes. We also present new results on the minimum distance of linear cyclic codes using their fixed subcodes. Finally, we list many new record-breaking quantum codes obtained from our constructions.
We introduce some methods for constructing quaternary Hermitian self-orthogonal (HSO) codes, and construct quaternary [n, 5] HSO for 342≤n≤492. Furthermore, we present methods of constructing Hermitian linear complementary dual (HLCD) codes from known HSO codes, and obtain many HLCD codes with good parameters. As an application, 31 classes of entanglement-assisted quantum error correction codes (EAQECCs) with maximal entanglement can be obtained from these HLCD codes. These new EAQECCs have better parameters than those in the literature.
A simple construction of quaternary hermitian self-orthogonal codes with parameters [2n+1,k+1][2n+1,k+1][2n+1,k+1] and [2n+2,k+2][2n+2,k+2][2n+2,k+2] from a given pair of self-orthogonal [n,k][n,k][n,k] codes, and its link to quantum codes is considered. As an application, an optimal quaternary linear [28,20,6][28,20,6][28,20,6] dual containing code is found that yields a new optimal [[28,12,6]][[28,12,6]][[28,12,6]] quantum code.
The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions on the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As a consequence, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, an entanglement-assisted quantum error-correcting [[21s + 5, 3, 16s + 3; 21s + 2]] code with maximal entanglement is constructed for the first time from a quaternary Hermitian linear complementary dual [26, 3, 19] code.
In this paper, we introduce a class of dualcontaining 2 -generator quasi-cyclic codes over GF(4)G F(4)GF(4). By Hermitian construction, we show computation constructions of binary quantum codes with good parameters from quaternary quasi-cyclic codes. Furthermore, 3 new quantum codes [[42, 10, 9]], [[74,,36, 9]] and [[78, 40, 9]] are derived from Hermitian dual-containing quasi-cyclic codes by an exhaustive search, along with 24 newly derived codes that enhance the previously best-known lower bounds on minimum distance.