Coset codes. II. Binary lattices and related codes

The family of Barnes-Wall lattices and their principal sublattices, which are useful in constructing coset codes, are generated by iteration of a simple construction called the squaring construction, and are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum-likelihood decoding algorithms.

Abstract:

For pt.I see ibid., vol.34, no.5, p.1123-51 (1988). The family of Barnes-Wall lattices (including D/sub 4/ and E/sub 8/) of lengths N=2/sup n/ and their principal sublattices, which are useful in constructing coset codes, are generated by iteration of a simple construction called the squaring construction. The closely related Reed-Muller codes are generated by the same construction. The principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions, which also exhibit the interrelationships between codes and lattices of different lengths. An extension called the cubing construction generates good codes and lattices of lengths N=3*2/sup n/, including the Golay code and Leech lattice, with the use of special bases for 8-space. Another related construction generates the Nordstrom-Robinson code and an analogous 16-dimensional nonlattice packing. These constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum-likelihood decoding algorithms. >

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Closest point search in lattices

Erik Agrell,Thomas Eriksson,Alexander Vardy,Kenneth Zeger +3 moreChalmers University of Technology,University of California, San Diego

- 01 Aug 2002 -

IEEE Transactions on Information Theory

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Mohamed Oussama Damen,H. El Gamal,Giuseppe Caire +2 moreUniversity of Alberta,Ohio State University,Institut Eurécom

- 01 Oct 2003 -

IEEE Transactions on Information Theory

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Niclas Wiberg

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Journal ArticleDOI

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- 01 Jan 1978 -

IEEE Transactions on Information Theory

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TL;DR: It is shown that soft decision maximum likelihood decoding of any (n,k) linear block code over GF(q) can be accomplished using the Viterbi algorithm applied to a trellis with no more than q^{(n-k)} states.

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