Affine Reflection Group Codes

Abstract

This correspondence presents a construction of affine reflection group codes. The solution to the initial vector and nearest distance problem is presented for all irreducible affine reflection groups of rank n ges 2, for varying stabilizer subgroups. We use a detailed analysis of the geometry of affine reflection groups to produce a decoding algorithm which is equivalent to the maximum-likelihood decoder, yet whose 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.

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Affine Reflection Group Codes
Terasan Niyomsataya1, Ali Miri1,2and 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
1Introduction
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 Rnwith 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
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orthogonal n × n matrix Ogfor 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) ∈ Rnis called the initial vector. The initial vector problem
is to choose a vector x0which 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 x0by 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 x0as 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 x0is 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, G2and A3in 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.
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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 A2and B2in 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
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r in Rn. For each root r of G and each integer k ∈ Z, the affine hyperplane Hr,kis defined as
Hr,k= {x ∈ Rn: r · x = k}
(2)
where r · x denotes the dot product of the two vectors. The affine reflection in this hyperplane is
denoted Sr,k∈ G; its action on a vector x ∈ Rnis defined by [7]
Sr,kx = x − (x · r − k) ·
= Srx +2kr
2r
r · r
(3)
r · r.(4)
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?G for the corresponding affine reflection group; but we omit the?, for simplicity.
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
Φ = Φ+∪ −(Φ+). The highest long root of G is a positive root ρ =?n
Throughout this paper, let us denote the set of all roots of G by Φ; they are generated by a root
i=1airiwhich is defined by the
property that the sum of its coordinates?n
roots.
i=1aiwith respect to the basis Π is maximal among all
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
obtained from [6, §12.1, Table 12.2]. We use {e1,e2,...,ek} to denote the standard basis of Rk. 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, E6and E7are
given so that each span a proper subspace of their ambient space of definition; this is convention. See
the Appendix for further discussion.
Tables of simple roots Π, of all roots Φ, and of the cardinalities of the corresponding irreducible
Weyl groups can also be found in [1, Table 5.1 and 5.2] or in [10, Table I and II].
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−2r1+ r2
−r1− r2
−r2
−r1
r1+ r2
2r1+ r2
r2
r1
Figure 1: The set Φ of all roots of B2
GroupSimple roots ΠHighest long root ρ
An
Bn
ri=
1
√2(ei+1− ei), 1 ≤ i ≤ n
r1= e1,
r1+ r2+ ··· + rn
2r1+ 2r2+ ··· + 2rn−1+ rn
ri= ei− ei−1, 2 ≤ i ≤ n
r1= e1+ e2,
Dn
r1+ r2+ ··· + 2rn−2+ 2rn−1+ rn
ri= ei− ei−1, 2 ≤ i ≤ n, n ≥ 4
r1=1
2
E6
?e1+ e8−?7
ri= ei−1− ei−2, 3 ≤ i ≤ 6
r1,...,r6, r7= e6− e5
r1,...,r7, r8= e7− e6
r1= e2− e3, r2= e3− e4, r3= e4
r4=1
2
i=2ei
?, r2= e1+ e2
r1+ 2r2+ 2r3+ 3r4+ 2r5+ r6
E7
E8
F4
2r1+ 2r2+ 3r3+ 4r4+ 3r5+ 2r6+ r7
2r1+ 3r2+ 4r3+ 6r4+ 5r5+ 4r6+ 3r7+ 2r8
2r1+ 3r2+ 4r3+ 2r4
2e1−1
?4
i=2ei
G2
r1= (−1,0),r2=
?
−3
2,
√3
2
?
3r1+ 2r2
Table 1: The simple roots and the highest long roots of the irreducible Weyl groups [5, 6, 10]
3The Initial Vector and Nearest Distance Problem
For a given irreducible affine reflection group G of rank n with simple roots Π = {r1,r2,...,rn} and
highest long root ρ =?n
domain D as
i=1airi, (where ai> 0 for each i ∈ {1,2,...,n}, we define the fundamental
D = {x ∈ Rn: x · ri> 0 for all ri∈ Π, and x · ρ < 1}.(5)
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