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IEEE Transactions on Information Theory

Quantum Cyclic and Constacyclic Codes

L. Xiaoyan

Published Mar 1, 2004• 44 citations
Publisher

Abstract

Based on classical quaternary constacyclic linear codes, we construct a set of quantum codes with parameters [[(4/sup m/ -1)/3, (4/sup m/ -1)/3 -2(3l + b)m, 4l + b + 2]] where m/spl ges/4, 1/spl les/b/spl les/3, and 12l + 3b < 2 /spl times/ 4/sup /spl lfloor/(m+2)/3/spl rfloor//-1, which are better than the codes in Bierbrauer and Edel (2000).

DOI
10.1109/tit.2004.825502

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Based on classical quaternary constacyclic linear codes, we construct a set of quantum codes with parameters [[(4/sup m/ -1)/3, (4/sup m/ -1)/3 -2(3l + b)m, 4l + b + 2]] where m/spl ges/4, 1/spl les/b/spl les/3, and 12l + 3b < 2 /spl times/ 4/sup /spl lfloor/(m+2)/3/spl rfloor//-1, which are better than the codes in Bierbrauer and Edel (2000).
Constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable (MDS) codes. However, the mathematical mechanism may not have been fully understood. In this paper, we use pseudo-cyclic codes, which is a further generalization of constacyclic codes, to construct the quantum MDS codes. We can not only provide a unified explanation of many previous constructions, but also produce some new quantum MDS codes.
Assume that qqq is a prime power and m≥2m\geq 2m≥2 is a positive integer. Cyclic codes over Fq2m\mathbb{F}_{q^{2m}}Fq2m​ of length n=q2m−1ρn=\frac{q^{2m}-1}{\rho }n=ρq2m−1​ with ρ∣(q−1)\rho\mid (q-1)ρ∣(q−1), and constacyclic codes over Fq2m\mathbb{F}_{q^{2m}}Fq2m​ of length n=q2m−1ρn=\frac{q^{2m}-1}{\rho }n=ρq2m−1​ with ρ∣(q+1)\rho\mid (q+1)ρ∣(q+1) are considered in this paper, respectively. Two classes of quantum codes are derived from the images of these codes by the Hermitian construction. Compared with the previously known quantum codes, the quantum codes in our scheme have better parameters.
If a linear block code C of length n has a Tanner graph without cycles, then maximum-likelihood soft-decision decoding of C can be achieved in time O(n/sup 2/). However, we show that cycle-free Tanner graphs cannot support good codes. Specifically, let C be an (n,k,d) linear code of rate R=k/n that can be represented by a Tanner graph without cycles. We prove that if R/spl ges/0.5 then d/spl les/2, while if R<0.5 then C is obtained from a code of rate /spl ges/0.5 and distance /spl les/2 by simply repeating certain symbols. In the latter case, we prove that d/spl les/[n/k+1]+[n+1/k+1]<2/R. Furthermore, we show by means of an explicit construction that this bound is tight for all values of n and k. We also prove that binary codes which have cycle-free Tanner graphs belong to the class of graph-theoretic codes, known as cut-set codes of a graph. Finally, we discuss the asymptotics for Tanner graphs with cycles, and present a number of open problems for future research.