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Families of Cyclic Codes over Finite Chain Rings

Andrew Nemec

Published Sep 29, 2015• 0 citations
OpenAlex

Abstract

A major difficulty in quantum computation and communication is preventing and correcting errors in the quantum bits. Most of the research in this area has focused on stabilizer codes derived from self-orthogonal cyclic error-correcting codes over finite fields. Our goal is to develop a similar theory for self-orthogonal cyclic codes over the class of finite chain rings which have been proven to also produce stabilizer codes. We also will discuss these restrictions on families of cyclic codes, including, but not limited to quadratic residue codes and Bose-Chaudhuri-Hocquenghem codes. Finally, we will extend the concepts of weight enumerators to the class of Frobenius rings and use them to derive bounds for our codes.

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A major difficulty in quantum computation and communication is preventing and correcting errors in the quantum bits. Most of the research in this area has focused on stabilizer codes derived from self-orthogonal cyclic error-correcting codes over finite fields. Our goal is to develop a similar theory for self-orthogonal cyclic codes over the class of finite chain rings which have been proven to also produce stabilizer codes. We also will discuss these restrictions on families of cyclic codes, including, but not limited to quadratic residue codes and Bose-Chaudhuri-Hocquenghem codes. Finally, we will extend the concepts of weight enumerators to the class of Frobenius rings and use them to derive bounds for our codes.
It is believed that quantum computers would be able to solve complex problems more quickly than any other deterministic or probabilistic computer. Quantum computers basically exploit the rules of quantum mechanics for speeding up computations. However, building a quantum computer remains a daunting task. A quantum computer, as in any quantum mechanical system, is susceptible to decohorence of quantum bits resulting from interaction of the stored information with the environment. Error correction is then required to restore a quantum bit, which has changed due to interaction with external state, to a previous non-erroneous state in the coding subspace. Until now the methods for quantum error correction were mostly based on stabilizer codes over finite fields. The aim of this thesis is to construct quantum error correcting codes over finite Frobenius rings. We introduce stabilizer codes over quadratic algebra, which allows one to use the hamming distance rather than some less known notion of distance. We also develop propagation rules to build new codes from existing codes. Non binary codes have been realized as a gray image of linear Z4 code, hence the most natural class of ring that is suitable for coding theory is given by finite Frobenius rings as it allow to formulate the dual code similar to finite fields. At the end we show some examples of code construction along with various results of quantum codes over finite Frobenius rings, especially codes over Zm.
One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes overbfFqbf F_qbfFq​in terms of classical codes over$ bf F_q^2isprovidedthatgeneralizesthewell−knownnotionofadditivecodesoveris provided that generalizes the well-known notion of additive codes overisprovidedthatgeneralizesthewell−knownnotionofadditivecodesover bf F_4$of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose–Chaudhuri–Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.
In this paper, we investigate the Gray images of codes over chain rings, leading to the derivation of infinite families of self-orthogonal linear codes over the residue field Fq\mathbb{F}_qFq​. We determine the parameters of optimal self-orthogonal and divisible linear codes. Additionally, we study the Gray images of quasi-twisted codes, resulting in some self-orthogonal Griesmer quasi-cyclic codes. Finally, we employ the CSS construction to derive some quantum codes based on self-orthogonal linear codes. Received: 19 September 2023 | Accepted: 18 January 2024
The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding classical self-orthogonal linear codes over the Galois field GF(4) under a Hermitian inner product.A classical cyclic code over the field GF(4) is self-orthogonal with respect to the Hermitian inner product iff the generator polynomial of its dual code is a factor of its generator polynomial.Based on this connection,a class of quantum cyclic codes is constructed by finding corresponding classical cyclic codes.Some examples are also given in detail.