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International Journal of Computer Mathematics Computer Systems Theory

Quantum codes from Hermitian dual-containing cyclic codes

Jiafu Mi, Xiwang Cao, Shanding Xu, Gaojun Luo

Published May 13, 2017• 4 citations
Publisher

Abstract

A new criterion for the existence of Hermitian dual-containing cyclic codes is obtained based on a characterization of q-cyclotomic cosets modulo n. This criterion admits a method of constructing new Hermitian dual-containing cyclic codes (hence new quantum codes) from the old ones, and enables us to construct several classes of quantum maximum-distance-separable (MDS) codes from classical Hermitian dual-containing cyclic codes. We analyse that our codes are good in certain sense. Besides, some enumerative conclusions are also presented as a result of the characterization of q-cyclotomic cosets modulo n.

DOI
10.1080/23799927.2017.1330284

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A new criterion for the existence of Hermitian dual-containing cyclic codes is obtained based on a characterization of q-cyclotomic cosets modulo n. This criterion admits a method of constructing new Hermitian dual-containing cyclic codes (hence new quantum codes) from the old ones, and enables us to construct several classes of quantum maximum-distance-separable (MDS) codes from classical Hermitian dual-containing cyclic codes. We analyse that our codes are good in certain sense. Besides, some enumerative conclusions are also presented as a result of the characterization of q-cyclotomic cosets modulo n.
In this paper, we use q -cyclotomic cosets modulo n = q − 1 and generator polynomial to describe repeated-root cyclic codes of length N = p α n . We determine the condition under which a repeated-root cyclic code contains its dual code. Then, for q = 2 a , we give a class of dual-containing repeated-root cyclic codes. Applying CSS construction and Steane construction to these underlying classical codes, many quantum codes with good parameters shall be constructed, which are new in the sense that they are not available in the literature.
Quantum maximum-distance-separable (MDS) codes form an important class of quantum codes. To get q-ary quantum MDS codes, one of the effective ways is to find linear MDS codes C over Fq2satisfying C⊥H⊆ C, where C⊥Hdenotes the Hermitian dual code of C. For a linear code C of length n over Fq2, we say that C is a dual-containing code if C⊥H⊆ C and C≠ Fq2n. Several classes of new quantum MDS codes with relatively large minimum distance have been produced through dual-containing constacyclic MDS codes. These works motivate us to make a careful study on the existence conditions for dual-containing constacyclic codes. We obtain necessary and sufficient conditions for the existence of dual-containing constacyclic codes. Four classes of dual-containing constacyclic MDS codes are constructed and their parameters are computed. Consequently, the quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have minimum distance bigger than the ones available in the literature.
Obtaining quantum maximum distance separable (MDS) codes from dual containing classical constacyclic codes using Hermitian construction have paved a path to undertake the challenges related to such constructions. Using the same technique, some new parameters of quantum MDS codes have been constructed here. One set of parameters obtained in this paper has achieved much larger distance than work done earlier. The remaining constructed parameters of quantum MDS codes have large minimum distance and were not explored yet.
A method of describing quaternary cyclic codes with 4-cyclotomic cosets of modulo odd n and generator polynomials is presented.A necessary and sufficient condition under which a quaternary cyclic code containing its dual code is given,thus generalizing the related known results on self-orthogonal simple root cyclic codes and on BCH codes to al quaternary cyclic codes.Using the relation of simple root cyclic codes and repeat root cyclic codes,the linear quantum codes that can be constructed from quaternary cyclic codes of short lengths are determined.
After the pioneering work of Shor and Steane, we are able to establish links between quantum codes and classical codes with certain self-orthogonality. Therefore, constructing classical self-orthogonal codes with small dimension and large dual distance becomes our centre point due to the interesting application to quantum codes. \n \n \n \nOne of the most interesting and useful families of classical codes are MDS codes. We will show a systematic construction for classical Hermitian self-orthogonal MDS codes through generalized Reed-Solomn codes. Afterwards, new families of quantum MDS codes can be produced. \n \n \n \nClassical BCH codes would be another good choice for self-orthogonal codes. We show the dual containment through polynomial evaluations, simply by choosing suitable cyclotomic cosets. As a result, quantum codes with good parameters can be derived. \n \n \n \nAnother good candidate of classical codes to investigate for dual containment comes from the class of \n \nalgebraic geometry codes. Instead of testing for dual containment directly, this thesis explores an equivalent condition for the existence of self-orthogonal AG codes. \n \n \n \nAfter various constructions for quantum codes, this thesis ends with a study on bounds for estimating the parameters of codes.