IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Some Improved Constructions for Nonbinary Quantum BCH Codes

Nianqi Tang, Zhuo Li, Lijuan Xing, Ming Zhang, Feifei Zhao

Published Dec 31, 2018• 5 citations

Abstract

Maximal designed distances for nonbinary narrow-sense quantum Bose-Chaudhuri-Hocquenghem (BCH) codes of length $n=\frac{q^4-1}{r}$ and new constructions for them are given, where q is an odd prime power. These constructions are capable of designing quantum BCH codes with new parameters. Furthermore, some codes obtained here have better parameters than those constructed by other known constructions.

DOI: 10.1587/transfun.e102.a.303

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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Some Improved Constructions for Nonbinary Quantum BCH Codes

Nianqi Tang, Zhuo Li, Lijuan Xing, Ming Zhang, Feifei Zhao

Published Dec 31, 2018• 5 citations

Publisher

Abstract

Maximal designed distances for nonbinary narrow-sense quantum Bose-Chaudhuri-Hocquenghem (BCH) codes of length

n=\frac{q^4-1}{r}

and new constructions for them are given, where q is an odd prime power. These constructions are capable of designing quantum BCH codes with new parameters. Furthermore, some codes obtained here have better parameters than those constructed by other known constructions.

DOI: 10.1587/transfun.e102.a.303

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Maximal designed distances for nonbinary narrow-sense quantum Bose-Chaudhuri-Hocquenghem (BCH) codes of length

n=\frac{q^4-1}{r}

q

-ary ($q\ensuremath{\ne}2$ is a prime power) Steane's enlargement of Calderbank-Shor-Steane codes applied to Euclidean self-orthogonal non-narrow-sense BCH codes. The quantum nonbinary BCH codes presented here have parameters better than the ones available in the literature.

Let

q

be a prime power and

m\geq 3

be odd. Suppose that

n=\frac {q^{2m}-1}{a}

with

a|(q^{m}+1)

and

3\leq a \leq 2(q^{2}-q+1)

. This paper mainly determines the actual maximum designed distance of Hermitian dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes over

\mathbb {F}_{q^{2}}

of length

n

. Firstly, we give the maximum designed distance

\delta _{m,a}^{R}

of narrow-sense Hermitian dual-containing BCH codes. Secondly, we show that there are also non-narrow-sense ones of designed distance up to

\delta _{m,a}^{R}

. It is worth mentioning that our maximum designed distance

\delta _{m,a}^{R}>\lceil \frac {a}{2}\rceil \delta _{m}^{A}

, where

\delta _{m}^{A}

is given by Alyet al.(IEEE Trans. Inf. Theory, vol. 53, no. 3, pp. 1183-1188, 2007). Thus, many families of Hermitian dual-containing BCH codes with relatively large designed distance are obtained. Using the Hermitian construction to them, we can subsequently construct different classes of nonprimitive quantum codes, which are new in the sense that their parameters are not covered in the literature.