International Congress on Image and Signal Processing (CISP)

A Class of Negacyclic Codes and Their Application to Quantum Codes

Junli Wang, Ruihu Li, Yang Liu, Hao Song

Published Oct 1, 2019• 0 citations

Abstract

In this paper, the properties of q2-ary cyclotomic cosets modulo 2n are investigated in detail, where n=[(q2m-1)/(m(q-1))],q ≡ ±1mod m or q2≡ -1mod m when m is even and q is an odd power prime when m is power of 2. Then, non-narrow-sense negacyclic Bose-Chaudhuri-Hocquenghem (BCH) codes of length n that contain their Hermitian duals are determined. By the Hermitian construction, series of quantum codes are obtained from such negacyclic BCH codes. Compared with quantum codes in the [6], [10], [12], our constructed ones have better parameters.

DOI: 10.1109/cisp-bmei48845.2019.8965687

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International Congress on Image and Signal Processing (CISP)

A Class of Negacyclic Codes and Their Application to Quantum Codes

Junli Wang, Ruihu Li, Yang Liu, Hao Song

Published Oct 1, 2019• 0 citations

Publisher

Abstract

DOI: 10.1109/cisp-bmei48845.2019.8965687

Related papers

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The Bose–Chaudhuri–Hocquenghem (BCH) codes have been studied for more than 57 years and have found wide application in classical communication system and quantum information theory. In this paper, we study the construction of quantum codes from a family of [Formula: see text]-ary BCH codes with length [Formula: see text] (also called antiprimitive BCH codes in the literature), where [Formula: see text] is a power of 2 and [Formula: see text]. By a detailed analysis of some useful properties about [Formula: see text]-ary cyclotomic cosets modulo [Formula: see text], Hermitian dual-containing conditions for a family of non-narrow-sense antiprimitive BCH codes are presented, which are similar to those of [Formula: see text]-ary primitive BCH codes. Consequently, via Hermitian Construction, a family of new quantum codes can be derived from these dual-containing BCH codes. Some of these new antiprimitive quantum BCH codes are comparable with those derived from primitive BCH codes.The Bose–Chaudhuri–Hocquenghem (BCH) codes have been studied for more than 57 years and have found wide application in classical communication system and quantum information theory. In this paper, we study the construction of quantum codes from a family of [Formula: see text]-ary BCH codes with length [Formula: see text] (also called antiprimitive BCH codes in the literature), where [Formula: see text] is a power of 2 and [Formula: see text]. By a detailed analysis of some useful properties about [Formula: see text]-ary cyclotomic cosets modulo [Formula: see text], Hermitian dual-containing conditions for a family of non-narrow-sense antiprimitive BCH codes are presented, which are similar to those of [Formula: see text]-ary primitive BCH codes. Consequently, via Hermitian Construction, a family of new quantum codes can be derived from these dual-containing BCH codes. Some of these new antiprimitive quantum BCH codes are comparable with those derived from primitive BCH codes.

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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.