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IEEE Transactions on Information Theory

Quantum Codes Derived From Certain Classes of Polynomials

Tao Zhang, Gennian Ge

Published Sep 21, 2016• 14 citations
Publisher

Abstract

One central theme in quantum error-correction is to construct quantum codes that have a relatively large minimum distance. In this paper, we first present a construction of classical linear codes based on certain classes of polynomials. Through these classical linear codes, we are able to obtain some new quantum codes. It turns out that some of the quantum codes exhibited here have better parameters than the ones available in the literature.

DOI
10.1109/tit.2016.2612578

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One central theme in quantum error-correction is to construct quantum codes that have a relatively large minimum distance. In this paper, we first present a construction of classical linear codes based on certain classes of polynomials. Through these classical linear codes, we are able to obtain some new quantum codes. It turns out that some of the quantum codes exhibited here have better parameters than the ones available in the literature.
One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. In this paper, we first present a construction of classical codes based on certain class of polynomials. Through these classical codes, we are able to obtain some new quantum codes. It turns out that some of quantum codes exhibited here have better parameters than the ones available in the literature. Meanwhile, we give a new class of quantum synchronizable codes with highest possible tolerance against misalignment from duadic codes.
The construction of quantum error-correcting codes has been an active field of quantum information theory since the publication of \cite{Shor1995Scheme,Steane1998Enlargement,Laflamme1996Perfect}. It is becoming more and more difficult to construct some new quantum MDS codes with large minimum distance. In this paper, based on the approach developed in the paper \cite{NewHeMDS2016}, we construct several new classes of quantum MDS codes. The quantum MDS codes exhibited here have not been constructed before and the distance parameters are bigger than q/2.
There have been various constructions of classical codes from polynomial valuations in literature [2], [7], [8], [10], [11]. In this paper, we present a construction of classical codes based on polynomial construction again. One of the features of this construction is that not only the classical codes arisen from the construction have good parameters, but also quantum codes with reasonably good parameters can be produced from these classical codes. In particular, some new quantum codes are constructed (see Examples V.5 and V.6).
One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. Quantum maximal distance separable (MDS) codes are optimal in the sense they attain maximal minimum distance. Recently, constructing quantum MDS codes has received much attention and seems to become more and more difficult. In this paper, based on classical constacyclic codes, we construct some new quantum MDS codes by employing the Hermitian construction. Compared with the known quantum MDS codes, these quantum MDS codes have much larger minimum distance.