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Affine Reflection Group Codes 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, [email protected] 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) n for all g ∈ G where x0 = (x1, . , xn) ∈ R 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 all codewords. It has not been solved for general groups, but rather is known only for special cases such as for cyclic groups [2], for the full homogeneous representation and the (n − 1)- dimensional representation of the symmetric groups of rank n [3] and for finite reflection groups [9, 10]. In this paper, we let G be an (infinite) affine reflection group of rank n ≥ 2, which has a natural action on Rn. The action of G restricted to any bounded portion of Rn (a codeword region) may be considered a Slepian group code, in the sense that: the group acts by isometries; the codewords are generated by multiplying an initial vector x0 by elements of the group; and, for a suitable choice of x0, they satisfy the nearest-neighbour equidistance property. Our main results are the solution to the initial vector problem, in Section 3, and the development of an efficient decoding algorithm for affine reflection group codes (bounded or not), in Section 5. The decoding algorithm is optimal, that is, equivalent to a maximum-likelihood decoder. We note that other group codes have provided alternatives to ML decoders. In particular, the length-reduction algorithm which is proposed in [9, Section IV] for finite reflection groups can be adapted directly for use with affine reflection group codes, but it would have high complexity for the case that the group code is large, or when the received vector lies very far from the fundamental domain. The paper is organized as follows. In Section 2 we give the mathematical background of affine reflection groups. In Section 3 we present the solution to the initial vector problem for any irreducible affine reflection group of rank n ≥ 2, that is, the method for computing the optimal choice of initial vector x0. Theorem 1 in this section gives a closed-form solution for x0 as well as a formula for the nearest-neighbour distance of the corresponding group code. Table 2 lists the initial vector and nearest distance of all affine reflection group codes. In Corollary 1, we solve the initial vector problem also for the case that the stabilizer subgroup of x0 is nontrivial. To illustrate our solution to the initial vector problem, and later to the decoding problem, we present the irreducible affine reflection groups A2, B2, G2 and A3 in detail. We include the necessary root data for all irreducible finite reflection groups in Table 1. In Table 2, we give the solutions to the initial vector problem for the corresponding irreducible affine reflection groups; the calculations have been relegated to the Appendix. 2 In Section 4 we define the bounded codeword regions R. These are designed to be maximally symmetric. We give examples of the size of these codes for irreducible affine reflection groups with bounded codeword regions in Table 3. An efficient decoding algorithm for the corresponding group code is proposed in Section 5. The complexity of the proposed decoding algorithm depends only the rank n of the affine reflection group (that is, on the dimension of the space spanned by the roots), and not on the size of the group code; this is in sharp contrast with the ML decoder. We summarize the algorithm briefly in Section 5.1 and Figure 4, showing how to efficiently reduce the decoding of a distant vector to one adjacent to the origin, where decoding techniques for finite reflection groups, such as the length-reduction algorithm of [9] may be applied effectively. We then proceed to describe the algorithm in detail, including proofs of correctness, in Section 5.2. The final step of the algorithm is illustrated in detail for the affine reflection groups A2, B2 and A3; in the first two of these we further optimize the algorithm by exploiting the many symmetries of the rank 2 groups. We then illustrate the complete algorithm with several examples from A2 and B2 in Section 6. We also give simulation results for a variety of cases, as further evidence of the correctness of the algorithm. Our conclusions are presented in Section 7. 2 Affine Reflection Groups An affine reflection group G (also known as an affine Weyl group) is a group generated by reflections in affine hyperplanes of Rn, and satisfying a local finiteness condition. Namely, if H is the set of all hyperplanes corresponding to reflections in G, then for any point in Rn, only finitely many hyperplanes in H intersect a neighbourhood of that point. We further say that G is irreducible if there does not exist a partition of H into mutually orthogonal sets; equivalently, if there does not exist a proper nonzero subspace which is preserved by all reflections in G. It is proven in [7, §4.10] that all irreducible affine reflection groups are generated by a finite, irreducible Weyl group (that is, a crystallographic finite reflection group), together with a single affine reflection. Hence these groups are completely classified by their Coxeter graphs; see for example [7]. All irreducible Weyl groups are listed in Table 1. More precisely: to each affine reflection group is associated a finite set of roots, which are vectors 3 n r in R . For each root r of G and each integer k ∈ Z, the affine hyperplane Hr,k is defined as n Hr,k = {x ∈ R : r · x = k} (2) where r · x denotes the dot product of the two vectors. The affine reflection in this hyperplane is n denoted Sr,k ∈ G; its action on a vector x ∈ R is defined by [7] 2r Sr,kx = x − (x · r − k) · (3) r · r 2kr = Srx + . (4) r · r The group G is generated by the set of all affine reflections Sr,k; it is infinite. From (3) and (4), we can see that the affine reflections with k = 0, Sr,0, are linear; the subgroup of G generated by these linear reflections is the corresponding Weyl group. If G is a Weyl group, one often writes Ge for the corresponding affine reflection group; but we omit the e, for simplicity. Throughout this paper, let us denote the set of all roots of G by Φ; they are generated by a root + base, which is a set of linearly independent simple roots Π = {r1,..., rn}. The set Φ of roots which are positive linear combinations of elements of Π are called the positive roots, and Π is such that + + Pn Φ = Φ ∪ −(Φ ). The highest long root of G is a positive root ρ = i=1 airi which is defined by the Pn property that the sum of its coordinates i=1 ai with respect to the basis Π is maximal among all roots. Then a minimal generating set for G is in fact {Sr,0 : r ∈ Π} ∪ {Sρ,1}. Example 1. Figure 1 shows all the roots of the irreducible affine reflection group B2. The root base is Π = {r1 = (1, 0), r2 = (−1, 1)} and the highest long root is ρ = 2r1 + r2 = (1, 1). In Table 1, we list the simple roots and the highest long roots of the irreducible Weyl groups, as k obtained from [6, §12.1, Table 12.2]. We use {e1, e2,..., ek} to denote the standard basis of R . For each Weyl group, the letter identifies its associated Coxeter graph and the subscript indicates the rank of the group, which is defined equivalently as the number of vertices in its Coxeter graph, and as the dimension of the vector space spanned by its simple roots. Note that the roots of An, E6 and E7 are given so that each span a proper subspace of their ambient space of definition; this is convention. See the Appendix for further discussion.
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