Affine Reflection Group Codes.pdf

Affine Reflection Group Codes Terasan Niyomsataya1, Ali Miri1,2 and Monica Nevins2 School of Information Technology and Engineering (SITE)1 Department of Mathematics and Statistics2 University of Ottawa, Ottawa, Canada K1N 6N5 email: {tniyomsa,samiri}@site.uottawa.ca, mnevins@uottawa.ca Abstract This paper presents a construction of Slepian group codes from affine reflection groups. The solution to the initial vector and nearest distance problem is presented for all irreducible affine reflection groups of rank n ≥ 2, for varying stabilizer subgroups. Moreover, we use a detailed analysis of the geometry of affine reflection groups to produce an efficient decoding algorithm which is equivalent to the maximum-likelihood decoder. Its complexity depends only on the dimension of the vector space containing the codewords, and not on the number of codewords. We give several examples of the decoding algorithm, both to demonstrate its correctness and to show how, in small rank cases, it may be further streamlined by exploiting additional symmetries of the group. 1 1 Introduction Slepian [11] introduced group codes whose codewords represent a finite set of signals combining coding and modulation, for the Gaussian channel. A thorough survey of group codes can be found in [8]. The codewords lie on a sphere in n?dimensional Euclidean space Rn with equal nearest-neighbour distances. This gives congruent maximum-likelihood (ML) decoding regions, and hence equal error probability, for all codewords. Given a group G with a representation (action) on Rn, that is, an 1Keywords: Group codes, initial vector problem, decoding schemes, affine reflection groups 1 orthogonal n× n matrix Og for each g ∈ G, a group code generated from G is given by the set of all cg = Ogx0 (1) for all g ∈ G where x0 = (x1, . . . , xn) ∈ Rn is called the initial vector. The initial vector problem is to choose a vector x0 which maximizes the Euclidean distance between a codeword and its nearest neighbour, over a