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A 2-adic approach to the analysis of cyclic codes

机译：2-adic方法分析循环码

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This paper describes how 2-adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over Z/sub 2(a)/, a/spl ges/2, the ring of integers modulo 2/sup a/. It provides a 2-adic proof of a theorem of McEliece that characterizes the possible Hamming weights that can appear in a binary cyclic code. A generalization of this theorem is derived that applies to cyclic codes over Z/sub 2(a)/ that are obtained from binary cyclic codes by a sequence of Hensel lifts. This generalization characterizes the number of times a residue modulo 2/sup a/ appears as a component of an arbitrary codeword in the cyclic code. The limit of the sequence of Hensel lifts is a universal code defined over the 2-adic integers. This code was first introduced by Calderbank and Sloane (1995), and is the main subject of this paper. Binary cyclic codes and cyclic codes over Z/sub 2(a)/ are obtained from these universal codes by reduction modulo some power of 2. A special case of particular interest is cyclic codes over Z/sub 4/ that are obtained from binary cyclic codes by means of a single Hensel lift. The binary images of such codes under the Gray isometry include the Kerdock, Preparata, and Delsart-Goethals codes. These are nonlinear binary codes that contain more codewords than any linear code presently known. Fundamental understanding of the composition of codewords in cyclic codes over Z/sub 4/ is central to the search for more families of optimal codes. This paper also constructs even unimodular lattices from the Hensel lift of extended binary cyclic codes that are self-dual with all Hamming weights divisible by 4. The Leech lattice arises in this way as do extremal lattices in dimensions 32 through 48.
机译：本文介绍了如何使用2-adic数来分析二进制循环码的结构以及在Z / sub 2（a）/，a / spl ges / 2，模2 / sup的整数环上定义的循环码的结构/。它提供了McEliece定理的2-adic证明，该定理描述了可能出现在二进制循环码中的汉明权重。得出该定理的一般化，适用于Z / sub 2（a）/上的循环码，该循环码是通过一系列Hensel提升从二进制循环码获得的。这种概括表征了残差模2 / sup a /作为循环码中任意码字的组成部分出现的次数。 Hensel提升序列的极限是在2 adic整数上定义的通用代码。该代码由Calderbank和Sloane（1995）首次引入，是本文的主要主题。通过减少2的幂模，从这些通用代码中获得Z / sub 2（a）/上的二进制循环码和循环码。一种特殊的特殊情况是从二进制循环中获得的Z / sub 4 /上的循环码。通过单个Hensel升降机进行编码。在格雷等轴测图中，此类代码的二进制图像包括Kerdock，Preparata和Delsart-Goethals代码。这些是比目前已知的任何线性代码包含更多代码字的非线性二进制代码。对Z / sub 4 /之上的循环码中的码字组成有基本的了解，对于寻找更多系列的最佳码至关重要。本文还从扩展二进制循环码的Hensel提升中构造出甚至是单模晶格，它们都是自对偶的，所有汉明权重都可以被4整除。Leech晶格以这种方式出现，尺寸为32至48的极值晶格也是如此。

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