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A 2-adic approach to the analysis of cyclic codes

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3 Author(s)
Calderbank, A.R. ; Inf. Sci. Center, AT&T Bell Labs., Murray Hill, NJ, USA ; Wen-Ching Winnie Li ; Poonen, B.

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 2(a), a⩾2, the ring of integers modulo 2a. 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 Z2(a) that are obtained from binary cyclic codes by a sequence of Hensel lifts. This generalization characterizes the number of times a residue modulo 2a 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 Z2(a) are obtained from these universal codes by reduction modulo some power of 2. A special case of particular interest is cyclic codes over Z4 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 Z4 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

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Information Theory, IEEE Transactions on  (Volume:43 ,  Issue: 3 )