The MacWilliams identities for nonlinear codes
The MacWilliams identities for nonlinear codes
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非线性代码的 MacWilliams 恒等式
DOI：
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发表时间：
1972
期刊：
影响因子：
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通讯作者：
J. Goethals
中科院分区：
文献类型：
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作者：
F. MacWilliams;N. Sloane;J. Goethals
近年来，许多非线性码被发现，它们比任何已知的线性码都具有更好的纠错能力。然而，人们对这种码的性质知之甚少。本文研究了最基本的性质--权枚举数。码字的权重是其非零分量的数量；权重枚举器给出每个权重的码字的数量，并且对于在噪声信道上使用代码进行纠错时获得差错概率是基本的。1963年，我们中的一个人证明了线性码的重量枚举器以一种简单的方式与对偶码的重量枚举器相关(Jessie MacWilliams，“关于系统码中重量的分布的定理”，贝尔系统技术期刊，42，第1期(1963年1月)，第79-94页)。在本文的续篇中，我们证明了对于非线性码的重量枚举子，同样的关系也成立。此外，还给出了满足(α⊥)的非线性二进制码α的对偶α⊥)⊥=α的定义，只要α包含零码字。
In recent years a number of nonlinear codes have been discovered which have better error-correcting capabilities than any known linear codes. However, very little is known about the properties of such codes. In this paper we study the most basic property, the weight enumerator. The weight of a codeword is the number of its nonzero components; the weight enumerator gives the number of codewords of each weight, and is fundamental for obtaining the error probability when the code is used for error-correction on a noisy channel. In 1963 one of us showed that the weight enumerator of a linear code is related in a simple way to that of the dual code (Jessie MacWilliams, “A Theorem on the Distribution of Weights in a Systematic Code,” Bell System Technical Journal, 42, No. 1 (January 1963), pp. 79–94). In the present paper, which is a sequel, we show that the same relationship holds for the weight enumerator of a nonlinear code. Furthermore, a definition is given for the dual α⊥ of a nonlinear binary code α which satisfies (α⊥)⊥ = α provided α contains the zero codeword.