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Nonbinary Stabilizer Codes Over Finite Fields

机译：有限域上的非二进制稳定器代码

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摘要

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 over Fq in terms of classical codes over Fq 2 is provided that generalizes the well-known notion of additive codes over F4 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
机译：量子通信和计算中的一个巨大困难是保护携带信息的量子态免遭与环境的有害相互作用。为了解决这个困难，许多好的量子纠错码已经被导出为二进制稳定器码。容错量子计算促进了对非二进制量子代码的研究，但是这种代码的理论并不像二进制量子代码那样先进。本文介绍了有限域上稳定器代码的基本理论。通过针对这些物体引入伽罗瓦理论来阐明稳定剂代码与一般量子代码之间的关系。提供了根据Fq 2上的经典代码来表征Fq上的非二进制稳定器代码，该特性概括了二进制情况下F4上的加性代码的众所周知的概念。本文还推导了稳定器代码最小距离的上限和下限，给出了几种代码构造，并得出了许多稳定器代码家族，包括量子汉明码，二次余数代码，量子梅拉斯代码，量子Bose-Chaudhuri-Hocquenghem（BCH） ）代码和量子字符代码。 Rains的穿孔理论被推广到不一定是纯净的附加代码。给出了最大距离可分离的稳定器代码的最大长度的界限。对开放问题的讨论总结了本文

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