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Journal of Air Force Engineering University

A Study of Skew Symmetric q~2-cyclotomic Coset and Its Application

Yang Liu

Published Jan 1, 2011• 19 citations
OpenAlex

Abstract

On the basis of concepts of skew symmetric-cyclotomic coset and skew asymmetric coset pair,and their properties deeply studied,and methods determined,a maximal design distance of BCH codes with length contained in its Hermitian dual is studied.Thus,such an unsolved problem for maximal design distance of BCH codes with length is solved,upper bound of maximal design distance of BCH codes previously known is improved,and a new upper bound picked up is sharp.By utilizing these Hermitian dual containing non-narrow-sense BCH codes,many new quantum codes with good parameters are constructed,these new quantum codes are better than that constructed from narrow-sense BCH codes in the literature.

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On the basis of concepts of skew symmetric-cyclotomic coset and skew asymmetric coset pair,and their properties deeply studied,and methods determined,a maximal design distance of BCH codes with length contained in its Hermitian dual is studied.Thus,such an unsolved problem for maximal design distance of BCH codes with length is solved,upper bound of maximal design distance of BCH codes previously known is improved,and a new upper bound picked up is sharp.By utilizing these Hermitian dual containing non-narrow-sense BCH codes,many new quantum codes with good parameters are constructed,these new quantum codes are better than that constructed from narrow-sense BCH codes in the literature.
Let q≥3q\geq 3q≥3 be a prime power. Maximal designed distances of primitive Hermitian dual containing q2q^{2}q2-ary BCH codes (narrow-sense or non-narrow-sense) are determined by a careful analysis of properties of cyclotomic cosets. Non-narrow-sense BCH codes which achieve these maximal designed distances are presented, and a sequence of nested non-narrow-sense BCH codes that contain these BCH codes with maximal designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these non-narrow-sense BCH codes. The nonbinary quantum BCH codes presented here have better parameters than those quantum BCH codes available in the literature.
Let q ≥ 3 be odd prime power. Maximal designed distances of non-primitive Hermitian dual containing q2- ary BCH codes of length n = (q4-1)/2 and 2(q2-1) are determined. For each given n, a group of non-narrow-sense Hermitian dual containing BCH codes are constructed and their dimensions are computed. Using these non-narrow-sense BCH codes, many new quantum codes are constructed. Our maximal designed distances improve upon the known results in the literature and many of new quantum BCH codes are better than those constructed from narrow-sense BCH codes.
By a careful analysis on cyclotomic cosets, the maximal designed distance δnew of narrow-sense imprimitive Euclidean dual containing q-ary BCH code of length [Formula: see text] is determined, where q is a prime power and l is odd. Our maximal designed distance δnew of dual containing narrow-sense BCH codes of length n improves upon the lower bound δmax for maximal designed distances of dual containing narrow-sense BCH codes given by Aly et al. [IEEE Trans. Inf. Theory53 (2007) 1183]. A series of non-narrow-sense dual containing BCH codes of length n, including the ones whose designed distances can achieve or exceed δnew, are given, and their dimensions are computed. Then new quantum BCH codes are constructed from these non-narrow-sense imprimitive BCH codes via Steane construction, and these new quantum codes are better than previous results in the literature.
This paper concentrates on construction of new nonbinary quantum error-correcting codes (QECCs) from three classes of narrow-sense imprimitive BCH codes over finite field [Formula: see text] ([Formula: see text] is an odd prime power). By a careful analysis on properties of cyclotomic cosets in defining set [Formula: see text] of these BCH codes, the improved maximal designed distance of these narrow-sense imprimitive Hermitian dual-containing BCH codes is determined to be much larger than the result given according to Aly et al. [S. A. Aly, A. Klappenecker and P. K. Sarvepalli, IEEE Trans. Inf. Theory 53, 1183 (2007)] for each different code length. Thus families of new nonbinary QECCs are constructed, and the newly obtained QECCs have larger distance than those in previous literature.
Let [Formula: see text] be an odd prime power and [Formula: see text] be a positive integer. Maximum designed distance such that negacyclic BCH codes over [Formula: see text] of length [Formula: see text] are Hermitian dual-containing codes is given. The dimension of such Hermitian dual-containing negacyclic codes is completely determined by analyzing cyclotomic cosets. Quantum negacyclic BCH codes of length [Formula: see text] are obtained by using Hermitian construction. The constructed quantum negacyclic BCH codes produce new quantum codes with parameters better than those obtained from quantum BCH codes.