Largest Minimum Weight Research Articles

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

We introduce quaternary modified four μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-circulant Hermitian self-dual codes. We also construct quaternary modified four μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-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.

Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes

We give two methods for constructing many linear complementary dual (LCD for short) codes from a given LCD code, by modifying some known methods for constructing self-dual codes. Using the methods, we construct binary LCD codes and quaternary Hermitian LCD codes, which improve the previously known lower bounds on the largest minimum weights.

Classification of optimal quaternary Hermitian LCD codes of dimension $2$

Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitian dual codes are trivial.
 The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$. Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitian dual codes are trivial.
 The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$.

Remark on subcodes of linear complementary dual codes

We show that any ternary Euclidean (resp. quaternary Hermitian) linear complementary dual [n,k] code contains a Euclidean (resp. Hermitian) linear complementary dual [n,k−1] subcode for 2≤k≤n. As a consequence, we derive a bound on the largest minimum weights among ternary Euclidean linear complementary dual codes and quaternary Hermitian linear complementary dual codes.

Some optimal entanglement-assisted quantum codes constructed from quaternary Hermitian linear complementary dual codes

We establish the existence of optimal entanglement-assisted quantum [Formula: see text] codes for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. These codes are obtained from quaternary Hermitian linear complementary dual codes. We also give some observations on the largest minimum weights.

Self-dual additive $ \mathbb{F}_4 $-codes of lengths up to 40 represented by circulant graphs

In this paper, we consider additive circulant graph codes which are self-dual additive \begin{document}$ \mathbb{F}_4 $\end{document} -codes. We classify all additive circulant graph codes of length \begin{document}$ n = 30, 31 $\end{document} and \begin{document}$ 34 \le n \le 40 $\end{document} having the largest minimum weight. We also classify bordered circulant graph codes of lengths up to 40 having the largest minimum weight.

Binary linear complementary dual codes

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study binary linear complementary dual $[n,k]$ codes with the largest minimum weight among all binary linear complementary dual $[n,k]$ codes. We characterize binary linear complementary dual codes with the largest minimum weight for small dimensions. A complete classification of binary linear complementary dual $[n,k]$ codes with the largest minimum weight is also given for $1 \le k \le n \le 16$.

New quantum codes constructed from some self-dual additive [formula omitted]-codes

For (n,d)=(66,17),(78,19) and (94,21), we construct quantum [[n,0,d]] codes which improve the previously known lower bounds on the largest minimum weights among quantum codes with these parameters. These codes are constructed from self-dual additive F4-codes based on pairs of circulant matrices.

There exists no self-dual [24,12,10] code over $${{\mathbb F}_5}$$

Self-dual codes over $${{\mathbb F}_5}$$ exist for all even lengths. The smallest length for which the largest minimum weight among self-dual codes has not been determined is 24, and the largest minimum weight is either 9 or 10. In this note, we show that there exists no self-dual [24, 12, 10] code over $${{\mathbb F}_5}$$ , using the classification of 24-dimensional odd unimodular lattices due to Borcherds.

On Binary Images of Shortened Cyclic (8, 5) Codes Over<tex>$rm GF(2^8)$</tex>

We have generated binary images of a large number of shortened cyclic (8, 5) codes over GF(2/sup 8/) and have computed weight distributions of the binary images of the codes. Based on the weight distributions, we have chosen four codes with the largest minimum weight 8 and the second largest minimum weight 7 among the generated codes. Over an additive white Gaussian noise channel with binary phase-shift keying modulation, simulation results have shown that block error rates of the chosen codes by a soft-decision decoding based on order-2 reprocessing are smaller than those of (64, 40) subcodes of Reed-Muller (64, 42) code by maximum likelihood decoding.