### Annals of Combinatorics 1 (1997) 279-295

Studies on the Squaring Construction

Kristian Wahlgren and Zhe-Xian Wan

Department of Information Technology, Information Theory Group, Lund University P.O. Box 118, S-221 00 Lund, Sweden
{kristian; wan}@it.lth.se

AMS Subject Classification: 05B40, 11H31, 94B75

Abstract. The code formulas for the iterated squaring construction for a finite group partition chain, derived by Forney [2], are extended to the case with a bi-infinite group partition chain, and the derivation presented here is much simpler than the one given by Forney for the finite case. It is also proven that the iterated squaring construction indeed generates the Reed-Muller codes. Moreover, the generalization of the code formulas to the bi-infinite case is used to derive code formulas for the lattices Λ (r,n) and R Λ (r,n), which correct some errors in [2].
Further, Gaussian integer lattices are discussed. A definition of their dual lattices is given, which is more general than the definition given by Forney [1]. Using this definition, some interesting properties of dual lattices and the squaring construction are obtained and then formulas of the duals of the Barnes-Wall lattices and their principal sublattices are derived, and one assumption from the derivation given by Forney [2] can be eliminated.

Keywords: lattice, squaring construction, Reed-Muller codes, Barnes-Wall lattice

References

1.  G.D. Forney, Jr., Coset codes - Part I: introduction and geometrical classification, IEEE Trans. Inform. Theory 34 (1988) 1123–1151.

2.  G.D. Forney, Jr., Coset codes - Part II: binary lattices and related codes, IEEE Trans. Inform. Theory 34 (1988) 1152–1187.

3.  F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam et al., 1977.

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