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IEEE International Symposium on Information Theory (ISIT)

On quantum MDS codes

Martin Rötteler, Markus Grassl, T. Beth

Published Dec 23, 2004• 55 citations
Publisher

Abstract

We construct maximum distance separable quantum error-correcting codes. The codes are defined over q-dimensional quantum systems, where q is any prime power. The construction yields quantum MDS codes of length up to q+1 for all possible dimensions and some quantum MDS codes of length up to q2+1. In particular, those families contain codes [[6,2,3]]pand [[7,3,3]]pfor pges3 (cf. [K. Feng (2002)])

DOI
10.1109/isit.2004.1365393

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We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q2, q2-2d+2, d〛q for 1≤d≤q which additionally give rise to shortened codes 〚q2-s, q2-2d+2-s, d〛q for some s.
It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. In particular, it is very hard to construct the quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known q-ary quantum MDS codes have minimum distance ≤q/2 + 1. In this paper, we provide a construction of the quantum MDS codes with minimum distance >q/2 + 1. In particular, we show the existence of the q-ary quantum MDS codes with length n = q2+ 1 and minimum distance d for any d q + 1 (this result extends those given in the works of Guardia (2011), Jin et al. (2010), and Kai an Zhu (2012)); and with length (q2+ 2)/3 and minimum distance d for any d (2q+2)/3 if 3|(q + 1). Our method is through Hermitian selforthogonal codes. The main idea of constructing the Hermitian self-orthogonal codes is based on the solvability in Fqof a system of homogenous equations over Fq2.
Abstract : We consider quantum error-correcting codes over alphabets of arbitrary size. We generalize quantum stabilizer codes and develop Calderbank-Shor-Steane construction over quantum systems of arbitrary dimensions using the group structure of alphabets. We also develop a methodology systematically conjoining error-correcting codes into a new class of error-correcting codes. With the help of these methods we give several ways to construct quantum maximum distance separable (MDS) codes and present many families of quantum MDS codes.
Entanglement-assisted quantum error-correcting codes, which can be seen as a generalization of quantum error-correcting codes, can be constructed from arbitrary classical linear codes by relaxing the self-orthogonality properties and using pre-shared entangled states between the sender and the receiver, and can also improve the performance of quantum error-correcting codes. In this paper, we construct some families of entanglement-assisted quantum maximum-distance-separable codes with parameters [[q2−1a,q2−1a−2d+2+c,d;c]]q[[\frac{{{q^2} - 1}}{a},\frac{{{q^2} - 1}}{a} - 2d+2 + c,d;c]]_q[[aq2−1​,aq2−1​−2d+2+c,d;c]]q​, where qqq is a prime power with the form q=am±ℓq = am \pm \ellq=am±ℓ, a=ℓ2−13a = \frac{{\ell^2} - 1}{3}a=3ℓ2−1​ is an odd integer, ℓ≡2 ( mod  6)\ell \equiv 2\ (\bmod\ 6)ℓ≡2 (mod 6) or ℓ≡4 ( mod  6)\ell \equiv 4\ (\bmod\ 6)ℓ≡4 (mod 6), and mmm is a positive integer. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature.