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Journal of Air Force Engineering University

Linear Quantum Codes Constructed from Quaternary Cyclic Codes

LI Rui-hu

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

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.

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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 letter, a method of constructing Hermitian dual containing quaternary linear codes from general quaternary linear codes is proposed. Based on this result, a construction method of binary linear quantum codes is obtained. Using three cyclic codes and one linear code over F4that are not Hermitian dual containing, we construct four record breaking binary linear quantum codes. Furthermore, ten new record breaking binary additive quantum codes are derived from these four linear quantum codes by propagation rule.
We present new constructions of binary quantum codes from quaternary linear Hermitian self-dual codes. Our main ingredients for these constructions are nearly self-orthogonal cyclic or duadic codes over F_4. An infinite family of 000-dimensional binary quantum codes is provided. We give minimum distance lower bounds for our quantum codes in terms of the minimum distance of their ingredient linear codes. We also present new results on the minimum distance of linear cyclic codes using their fixed subcodes. Finally, we list many new record-breaking quantum codes obtained from our constructions.