GROUP CODES WITH COMPLEX PERMUTATION GROUPS

Transcription

1 GROUP CODES WITH COMPLEX PERMUTATION GROUPS HYE JUNG KIM, J. B. NATION AND ANNE SHEPLER In memory of Wes Peterson. Abstract. Group codes that use certain complex matrix groups are analyzed. These groups are complex analogues of the real Coxeter reflection groups. It is shown that some types of complex permutation groups have good properties for use in coding, while others do not. 1. Introduction Permutation group codes originated in unpublished memos of David Slepian in the 1950 s. These codes are derived by choosing a point on an n-dimensional sphere and acting on it with a group of operations consisting of permutations of the coordinates and reversals of the signs of coordinates. This work was published in 1965 [1], and soon extended by Slepian to include other groups of isometries [2]. See the survey of Ingemarsson [3] and Ericson [4] for some earlier work on group codes. The generalizations of permutation group codes developed thus far have concentrated on using real reflection groups (Coxeter groups), which have a well-understood structure and action. The 1996 paper of Mittelholzer and Lahtonen [5] is particularly comprehensive. These algorithms were refined recently by Fossorier, Nation and Peterson [6]. Real reflection groups act on the unit sphere in a real vector space R n. Wes Peterson was always curious to know what other groups might have an action that lends itself well to coding. This note extends the group codes using the real permutation groups A n, B n, D n to codes based on complex permutation groups G(r,k,n) acting on the unit sphere in C n. Some care is necessary with the modifications involved, and this led us to carry out an analysis of the desirable properties for a group coding scheme. It turns out that the groups G(r,1,n) have good coding properties, while the groups G(r,k,n) with k > 1 do not. 2. Description of the algorithm The outline of a general group coding algorithm is straightforward. The setup involves selecting a particular group G of isometries acting on a vector Date: January 5,

2 2 HYE JUNG KIM, J. B. NATION AND ANNE SHEPLER space V. Then find an appropriate sequence of subgroups {I} = G 0 < G 1 < G 2 < < G m = G and (left) coset leaders for consecutive pairs of subgroups, G k over G k 1, with I as the coset leader for G k 1. Then every element of G has a unique expression as a product of coset leaders, g = c m...c 1, with each c k a coset leader for G k over G k 1. This may be regarded as a sort of canonical form for group elements. Now choose an initial vector x 0 on the unit sphere in V. Let us assume that the group G acts faithfully on x 0, though there are some interesting codes where this is not the case. The code consists of Gx 0 = {gx 0 : g G}. One should also establish a correspondence between messages and group elements, γ : M G. The details of this correspondence do not concern us here, and indeed we can just regard the canonical expression of an element g as its message. All the above things appear as subroutines or parameters in the implementation, which goes thusly. The message m to be sent corresponds to a group element g = γ(m). The element g has a canonical expression g = c m...c 1 as a product of coset leaders. The vector x = g 1 x 0 = c c 1 m x 0 is then transmitted. The received vector has the form r = x + n where n represents channel noise. Hopefully, r will be close to x, i.e., the (euclidean) distance r x 0 will be small. Now decode by iteratively finding the sequence of coset leaders d 1,...,d m such that, for each k, the element h k = d k... d 1 minimizes the distance hr x 0 over all h G k. (This amounts to a greedy algorithm, and it only works for certain carefully selected groups and subgroup sequences.) Thus the final element g = d m...d 1 minimizes the distance hr x 0 over all h G. Then decode by taking the received message as m = γ 1 (g ). As in the case of real reflection groups, the subgroups G k in our sequence will be determined by specifying a set of generators for each. Given a group G, a subgroup H G, and a set X of generators for G, the coset leader graph Γ for G over H is formed as follows. The vertices of Γ are the (left) coset leaders for G over H. Let us assume that the coset leaders are chosen so that (1) the identity I is a coset leader, and (2) if v I is a coset leader, then there exist a generator a X and a coset leader u such that v = au. In Γ, there is a directed edge from u to v whenever v = au for some generator a. Label such an edge by a, so that the transformation v can be reconstructed by tracing a path from I to v and reading the edge labels in order. Such a path need not be unique. However, as shown for example in [6], by ordering the generators it is possible to identify a canonical path from I to each coset leader v, and this determines a spanning tree T in Γ. For the subgroup sequences in permutation groups considered here, the coset leader graphs

3 GROUP CODES WITH COMPLEX PERMUTATION GROUPS 3 for consecutive pairs of subgroups will be trees or cycles, and issues that arise in more complicated groups play no role. (See Property 4 below.) 3. Necessary and desirable properties Let us catalogue the properties that the group, subgroup sequence and initial vector should have in order for the group coding algorithm to work and/or work well. Some of the nice features of real reflection groups are not automatic here, so we procede carefully Basic assumptions. Concerning the group: G is a (finite) group of isometries acting on R n or C n Parameters at our disposal. These can be adjusted. The subgroup sequence. The initial vector x 0 with x 0 = 1. The sequence is not essential for coding, but rather a basic source of efficiency. The subgroup sequence and initial vector will be subjected to various conditions below. These parameters are related to the subgroup sequence and initial vector. Generating sets X for G and Y k for G k (so Y m = X). Coset leaders CL k for G k over G k 1. Neighbors N k (x 0 ) = {hx 0 : I h G k and hx 0 x 0 gx 0 x 0 whenever I g G k }. Let N(x 0 ) = N m (x 0 ). The corresponding group elements N k = {h G k : hx 0 N k (x 0 )} and N = {h G : hx 0 N(x 0 )}. Coset leader graphs Γ k. Spanning trees T k for Γ k. A branch order on each T k. The last three pertain to the details of decoding, as in [6] Initial vector property. The choice of the initial vector determines a lot of things, including the neighbor sets N k (x 0 ). The minimum distance d min of a group code is min g g gx 0 g x 0, or equivalently min g I gx 0 x 0. Property 1. The group should act faithfully on the initial vector, and it should be chosen so as to make the minimum distance large. Group coding can be done with isotropy subgroups that fix x 0. Let us avoid that complication, for now at least. Note that it is not necessary to maximize the minimum distance in order for the decoding to work, but rather desirable that it be large for the code to work better. Part of the point of using complex unitary groups is that it allows a larger minimum distance for the same dimension. A later section will deal with some of the difficulties attending the choice of the initial vector.

4 4 HYE JUNG KIM, J. B. NATION AND ANNE SHEPLER 3.4. Coset leaders. Given the subgroup sequence, we must choose the coset leaders, the coset leader graphs and their spanning trees. When the subgroup sequence is given in terms of its generating sets, there is a standard way to do this, as in [6]. The only modification is that, if the generators are not involutions, then we must allow their inverses into the scheme. The next property has already been mentioned. Property 2. The coset leaders for consecutive subgroups in the subgroup sequence should be chosen so that, for each k, (1) the identity I is in CL k, and (2) if v I is a coset leader, then there exist a generator a Y k and u CL k such that v = au or v = a 1 u Navigation. A crucial part of the decoding algorithm is to navigate through the coset leader trees Γ i for G i over G i 1. The next property, written generically, is to be applied to consecutive terms G i 1 G i of the subgroup sequence. Property 3. Let H G and let x be a vector such that x x 0 < hx x 0 whenever I h H. Then there is an efficient way to find a coset leader c such that cx x 0 < dx x 0 for all coset leaders d c. Generally speaking, one moves through the coset leader tree downward from its root at I, each step getting closer to the initial vector, ceasing when further steps take you further away again. It is desirable in a group code that gx 0 x 0 be roughly proportional to the (minimum) length of g as a word in generators and their inverses. However, we leave open the possibility that there are ways to improve the process. For example, one can shortcut the process for rotations, where the coset leader graph is a cycle. So the next property is superfluous for permutation groups, but it was definitely needed for the real exceptional reflection group case. Otherwise, decoding is done incorrectly. So let s keep it in the list. Within the spanning trees, a branch order should be given on the edges eminating from a each vertex v, to determine the order in which the successors of v will be considered. The branch orders must satisfy a technical property that is crucial to the decoding algorithm. Property 4. Let u and w be vertices in the coset leader graph Γ. If there is a path in Γ going from u to w, then the (unique) path from u to w in the spanning tree T goes through the successor of u that is least in the branch order and lies on some path from u to w Subgroup decoding property. Temporarily ignore noise, and let us find a condition to ensure that a received vector sufficiently close to g 1 x 0 will be decoded correctly. Recall that the fundamental decoding region of G is given by FDR = {x : x x 0 < gx x 0 for all g I}.

5 GROUP CODES WITH COMPLEX PERMUTATION GROUPS 5 These are the vectors that would decode to I. The vectors that would decode to g G are DR(g) = {x : x g 1 x 0 < hx g 1 x 0 for all h I} which is easily seen to satisfy DR(g) = g 1 (FDR). Our decoding procedure is a greedy algorithm, in that we start with a received vector r, and for each subgroup in the sequence we find the coset leader c i that moves c i 1...c 1 r as close to the initial vector x 0 as possible. Greedy algorithms don t always work. In order to assure that the algorithm actually finds the group element g that minimizes gr x 0, each consecutive pair of the subgroup sequence should have the following property. Property 5. Let H G and let x be a vector such that x x 0 < hx x 0 whenever I h H. Let c be a coset leader such that cx x 0 < dx x 0 for all coset leaders d c. Then cx x 0 < gx x 0 for all g G with g c. The alternative, of course, is that some element dhx with h I could be closer. Our basic argument goes thusly. Lemma 1. Let H G and assume that Property 5 holds. Let x be a vector such that x x 0 < gx x 0 whenever I g G, i.e., x is in the fundamental decoding region of G. Then for a coset leader c of G over H, cx x 0 < chx x 0 whenever I h H. In particular, when Property 5 holds, coset leaders do indeed give the closest codewords in their coset: cx 0 x 0 < chx 0 x 0 whenever I h H. Proof. Let xbe as in the lemma. Starting with c 1 x, find h H so that hc 1 x x 0 is minimized over all choices of h. Set y = hc 1 x, and find the coset leader d that minimizes dy x 0. By the property applied to y, we must have that dy x 0 < gx x 0 for all g G, so that dy = dhc 1 x is in the fundamental decoding region. As G acts faithfully on x 0, that implies dhc 1 = I, or dh = c. Since c and d are both coset leaders, we must have c = d and h = 1, as desired. Now let us apply this argument recursively to show that the greedy decoding process works properly when the error is small. Theorem 2. Assume that each consecutive pair in the subgroup sequence {I} = G 0 < G 1 < G 2 < < G m = G satisfies Property 5. Let g G be written as a product of coset leaders, g = c m...c 1. Let x be a vector in the decoding region DR(g). Recursively find coset leaders d k for G k over G k 1 such that d k...d 1 x x 0 is minimized over all choices of the coset leader d k. Then g = d m...d 1, and hence d j = c j for all j.

6 6 HYE JUNG KIM, J. B. NATION AND ANNE SHEPLER Proof. Inductively, by Lemma 1, d k...d 1 x x 0 < d k...d 1 hx x 0 whenever I h G k. Thus d m...d 1 x is in the fundamental decoding region of G, so that x decodes as d m...d 1. On the other hand, x DR(g) means by definition that x = g 1 y for some y FDR. As G acts faithfully, this implies g = d m...d Error control. Now consider what happens when the received vector is not in the decoding region DR(g). The nearest neighbors of a codeword u are the codewords v with u v = d min. Lemma 3. The nearest neighbors of g 1 x 0 are the codewords g 1 w with w a nearest neighbor of x 0. Thus the nearest neighbors of g 1 x 0 will be of the form g 1 bx 0 = (bg) 1 x 0 with b N. If a codeword g 1 x 0 is decoded incorrectly, it will most likely be decoded as a neighbor (bg) 1 x 0 with b N. To minimize the message error, we would like the canonical form of bg to differ as little as possible from that of g. That is the impact of the next two desirable properties for consecutive subgroups in the subgroup sequence (from [6]). In the real case, we had X a fundamental set of reflections, so that X = N. More generally, the generators for each subgroup were taken to be a fundamental set of reflections for that subgroup, so that Y k = N k. (This was not explained as well as could be there.) Lemma 1 says that neighbors will come from coset leaders, but in fact they should be generators and their inverses (cf. Property 2). Property 6. For each k, N k Y k Y 1 k. Note that this property depends not only on the group, but on the choice of the initial vector x 0. To minimize the effect of small decoding errors, we want this property for consecutive pairs of subgroups. Property 7. Let H G with fixed generating sets X for G and Y for H, respectively. Let b X X 1 and let c be a left coset leader for G over H. Then the coset leader for bch is either bc, or else it is c and c 1 bc is in Y Y 1. Note that ch = c(c 1 bc)h. This minimizes the effect of small errors. Theorem 4. Suppose that Property 7 holds for the consecutive pairs of a sequence {I} = G 0 < G 1 < < G m 1 < G m = G. If the canonical form of an element g as a product of coset leaders is c m...c 1, and b X X 1 with X the generating set for G, then the canonical form of bg is c m... c 1 where c i = c i for all but one i, and for that index c j is an immediate predecessor or successor of c j in the coset leader graph Γ j for G j over G j 1.

7 GROUP CODES WITH COMPLEX PERMUTATION GROUPS 7 4. Problems with the initial vector Mittelholzer and Lahtonen [5] gave an elegant and simple solution to the initial vector problem in the real case. Any unit vector in the fundamental region can be taken for the initial vector; some work better than others, and there is a straightforward algorithm to find the optimal choice. The geometry is different in the complex case, and this introduces some complications. In this section, we explore some of the difficulties. The relevant considerations for choosing x 0 are given in Properties 1, 5, 6, and to a lesser extent Property 3. A natural approach would be choose x 0 so as to maximize the minimum distance d min, and then hope that Properties 5 and 6 hold. For any particular group, it is straightforward to write a program to approximate initial vectors maximizing d min. First, one should make the following observations. Lemma 5. Let x 0 be a vector that maximizes d min. (1) If c = 1, then cx 0 also achieves the maximum d min, with the same group elements N. (2) If h G, then hx 0 also achieves the maximum d min, with the group elements hgh 1 for g N. The first part means that the first entry of x 0 may be taken to be real (or imaginary), which can be useful. Our numerical experiments began with the groups G(r, 1, 2) for various values of r. Immediately, things started to go wrong. Remember that x 0 determines the neighbors N(x 0 ), thence the corresponding group elements N. With the groups G(r,1,2), for any choice of x 0 that maximizes d min, there are six elements in N. Moreover, this includes elements g, h such that g, h and gh are all in N. On the other hand, there is no choice of x 0 whose neighbors contain the natural generating set s 1, s 2, t 2. None of this is consistent with any reasonable interpretation of Properties 5 and 6. Maximum values of d min are given in Table 1 for some of the groups G(r, 1, n) considered in Section 6. Under the circumstances, these should be regarded as an upper bound, not attainable for good codes. For the purposes of this paper, we will assume that x 0 is a real unit vector, and adjust the entries to make neighbors of a preferred set of generators. Then we check Property 5, and compare the minimum distance to the known upper bound. In the long run, this problem needs more investigation. 5. Real permutation groups Let us review the real permutation groups that have been used in group coding. These are finite groups of orthogonal matrices acting on the unit sphere of a real finite-dimensional vector space. The group A n is the group of all (n + 1) (n + 1) permutation matrices. Clearly this is isomorphic to the symmetric group on n + 1 letters, as its

8 8 HYE JUNG KIM, J. B. NATION AND ANNE SHEPLER Table 1. Maximum d min for some G(r,1,n) r n = 2 n = 3 n = elements act on a vector in R n+1 by permuting its entries. The group A n acts irreducibly on the n-dimensional subspace {x R n+1 : x i = 0}, whence the notation. The group B n consists of all n n matrices with exactly one nonzero entry in each row and column, that entry being ±1. The elements of B n act on a vector in R n by permuting the entries and changing some of the signs. The group D n is the subgroup of B n consisting of those matrices in B n with an even number of 1 s. Its elements act on a vector in R n by permuting the entries and changing an even number of signs. 6. A concrete example In this section, we apply the above program to the complex reflection groups G(r,1,n). These groups are wreath products, extensions of (Z/rZ) n by the symmetric group S n. Let ρ = e 2πi r. The matrix representation of G(r,1,n) acting on the complex space C n consists of all matrices with one non-zero entry in each row and column, that entry being a power of ρ. In particular, G(2,1,n) is the real Coxeter group B n. The group coding scheme for B n extends very naturally to G(r,1,n) with minimal changes and a straightforward encoding/decoding scheme. For a (redundant) set of generators, take the transformations s i (1 i n) that multiply the i-th entry of a vector by ρ, and the transpositions t j (2 j n) that switch the j-th and (j 1)-st components. Consider the nested sequence of subgroups, and associated coset leaders, given by the setup in Table 2. At the even steps, G 2k is obtained by adding adding a generator s j that commutes with the elements of G 2k 1. At the odd steps, G 2k+1 is obtained by adding adding a generator t j, and the relations t j s j t j = s j 1 make all but say s 1 redundant. The coset leader graphs for G(4,1,4) are given in Figure 1. At this point, we can check Properties 2, 4 and 7. The first two clearly hold, though we need to discuss the cyclic coset leader graphs; see below.

9 GROUP CODES WITH COMPLEX PERMUTATION GROUPS 9 Table 2. Subgroup sequence for G(r, 1, n) k Generators for G k Coset leaders for G k over G k 1 0 I 1 s 1 s j 1 for 0 j < r 2 s 1,s 2 s j 2 for 0 j < r 3 s 1,t 2 I,t 2 4 s 1,t 2,s 3 s j 3 for 0 j < r 5 s 1,t 2,t 3 I,t 3,t 2 t n 1 s 1,t 2,...,t n I,t j...t n for 2 j n Figure 1. Coset leader graphs for G(4,1,4) s 1 s 1 s 2 s 2 s 3 s 3 s 4 s 4 t 2 t 3 t 4 s 1 s 1 s 2 s 2 s 3 s 3 s 4 s 4 t 2 t 3 t 2 Verifying the third property is mechanical, using these relations: s r j = 1 t 2 j = 1 s i s j = s j s i (t j 1 t j ) 3 = 1 s j 1 t j = t j s j t i t j = t j t i if i j > 1 s j t j = t j s j 1 s i t j = t j s i if i j,j 1. Next comes the task of choosing an appropriate initial vector. The goal is to make the neighbors of x 0 to be its images under the natural generating set s 1,t 2,...,t n. This can be done by mimicking the optimal vector for the real group B n. Assume that x 0 is a real vector of the form and require that x 0 = a,a + b,a + 2b,...,a + (n 1)b s 1 x 0 x 0 = t 2 x 0 x 0 = = t n x 0 x 0. A straightforward computation that leads to the condition b a = 1 cos 2π r with the minimum distance being 2b. Initially we set a = 1, and then normalize so that x 0 = 1. Note that s i x 0 x 0 will be greater for i > 1.

10 10 HYE JUNG KIM, J. B. NATION AND ANNE SHEPLER Table 3. Actual d min obtained for some G(r,1,n) r n = 2 n = 3 n = With regard to Property 1, the group acts faithfully on this x 0, and the minimum distances it yields have the right order of magnitude. For example, For G(4,1,2), d min.63 versus a maximum possible of.82. For G(6,1,2), d min.51 versus a maximum possible of.59. For G(8,1,2), d min.42 versus a maximum possible of.46. For G(4,1,3), d min.38 versus a maximum possible of.54. For G(4,1,4), d min.26 versus a maximum possible of.43. Table 3 gives the values achieved for small values of r and n. In our current situation, Property 6 says that if g I,s 1,s 1 1 or some t j, then gx 0 x 0 2 > 2b 2. This distance squared is the sum of terms of the form ρ t (a + jb) (a + lb) 2 = [c (a + jb) (a + lb)] 2 + [s (a + jb)] 2 where ρ t = c + is. We want to show that if j = l and t 0, then this expression is at least 2b 2, while if j l then it is at least b 2. Two separate but easy calculations complete the argument (using c Re(ρ) < 1). Notice that the fundamental regions depend on the choice of the initial vector x 0. This is different from the real case. Consider for example the first case with H = s 1. Writing x = (x 1,...,x n ) and x 0 = (u 1,...,u n ), we have FR( s 1 ) = {x C n : s k 1x x 0 > x x 0 for 1 k < r} = {x C n : Re(x 1 u 1 ) > Re(ρ k x 1 u 1 ) for 1 k < r}. This justifies in part our standard choice of x 0 as indicated above. Now consider Property 3. The fact that, in the complex case, some of the generators are not transpositions introduces a minor change in the algorithm. The generators s i satisfy s r i = 1, and in terms of the algorithm one should regard s r 1 i = s 1 i as a coset leader of length one, as it will move the initial vector by the same distance that s i does. This means that in decoding a vector x, at the appropriate stage one should consider x x 0, s i x x 0 and s 1 i x x 0. If the first distance is a minimum, you consider the next subgroup. If the minimum is one of the latter two, then proceed around the cycle in that direction until you reach a minimum, at most halfway around. The halfway point, however, could be approached from either direction.

11 GROUP CODES WITH COMPLEX PERMUTATION GROUPS 11 In fact, in the middle of the next argument, we find a better way to navigate these cyclic graphs. Finally, we must verify Property 5. This does not require the particular initial vector chosen above, but let us assume that x 0 = (u 1,...,u n ) with 0 < u 1 < < u n real. Consider a vector x = (x 1,...,x n ) C n. An easy calculation shows that, for x C and 0 < u R, the value of k that minimizes ρ k x u is the one that maximizes the real part of ρ k x, independently of the size of u. N.B. The preceding observation can be used to speed up the algorithm considerably. Writing x = x e iθ, then ρ k x = x e (2πk r +θ)i, the real part of which is maximized by making 2πk r + θ as closed to 2π as possible. Thus k should be chosen as the nearest integer to r rθ 2π. Now consider the sequence x x 0 s k 1x x 0 s l 2s k 1x x 0 t δ 2 sl 2 sk 1 x x 0 s m 3 t δ 2s l 2s k 1x x 0 cs m 3 tδ 2 sl 2 sk 1 x x 0 where c is a coset leader for G 5 over G 4, thus one of {I,t 3,t 2 t 3 }. First k is chosen to maximize Re(ρ k x 1 ), then l to maximize Re(ρ l x 2 ). Now since u 1 < u 2, an easy calculation shows that if Re(ρ k x 1 ) > Re(ρ l x 2 ), then we should apply s 2, switching the values, to minimize the distance; otherwise not. Next m is chosen to maximize Re(ρ m x 3 ). Then, since u 1 < u 2 < u 3, we apply the correct coset leader c to put Re(ρ k x 1 ), Re(ρ l x 2 ), Re(ρ m x 3 ) into increasing order (insertion sort). Continue until pau G(r,k,n) For each divisor k of n, there is a subgroup G(r,k,n) of G(r,1,n). Recall that, in terms of the matrix representation, G(r,1,n) consists of all n n matrices with one non-zero entry in each row and column, the entry in the i-th row being say ρ a i. The subgroup G(r,k,n) consists of all such matrices with n i=1 a i = 0 mod k. For example, G(2,2,n) is the real Coxeter group D n. For a (redundant) set of generators, take the transformations w i = s k i for 1 i n that multiply the i-th entry of a vector by ρ k, the transformations v i for 2 i n that multiply the i 1-st entry by ρ 1 and the i-th entry by ρ, and

12 12 HYE JUNG KIM, J. B. NATION AND ANNE SHEPLER Table 4. Subgroup sequence for G(r,k,n) k Generators for G k Coset leaders for G k over G k 1 0 I 1 w 1 w j 1 for 0 j < r k 2 w 1,w 2 w j 2 for 0 j < r k 3 w 1,v 2 v j 2 for 0 j < k 4 w 1,v 2,t 2 I,t 2 5 w 1,v 2,t 2,w 3 w j 3 for 0 j < r k 6 w 1,v 2,t 2,v 3 v j 3 for 0 j < k 7 w 1,v 2,t 2,t 3 I,t 3,t 2 t the transpositions t j for 2 j n that switch the j-th and (j 1)-st components. In the case k = r, we have w i = I for all i, so these may be omitted. Table 4 gives a natural subgroup sequence for G(r,k,n). The relations w i 1 v k i = w i and t j 1 t j v j 1 t j t j 1 = v j make the generators w i for i > 1 and v j for j > 2 redundant in later subgroups. Here are the relations on the generators for G(r,k,n). w r k j = I v r i = I t 2 j = I v i v j = v j v i (t j 1 t j ) 3 = 1 t i t j = t j t i if i j > 1 w i t i = t i w i 1 w i t j = t j w i if i j,j 1 v i t i = t i vi 1 w i v j = v j w i t i v i+1 = v i+1 v i t i t i+1 v i = v i+1 v i t i+1. The initial value problem is thorny, and we can only scratch the surface. To begin, there are a couple of natural options for the choice of neighbors, and one or the other may not work for a given r and k. The first option is to design x 0 so that its neighbors are its images under w 1,t 2,...,t n. Proceeding exactly as when k = 1, we set and require that x 0 = a,a + b,a + 2b,...,a + (n 1)b w 1 x 0 x 0 = t 2 x 0 x 0 = = t n x 0 x 0. The same calculation yields b a = 1 c

13 GROUP CODES WITH COMPLEX PERMUTATION GROUPS 13 where now c = Re(ρ k ) = cos 2kπ r. The distance gx 0 x 0 remains 2b for g either w 1 or some t j. On the other hand, v 2 x 0 x 0 2 = 2(1 c)(2a 2 +2ab 2 b ) where c = cos 2π r, and sometimes this is less than 2b 2, making v 2 x 0 and v2 1 x 0 instead the nearest neighbors of x 0. For example, this is the case whenever k = 2 and r 14 is even. In those situations, the second method is preferable. The second option is to design x 0 so that its neighbors correspond to v 2,t 2,...,t n. Again taking the same form for x 0, we find that, for r 5, b a = 1 c + 1 c 2. c For r = 3 use b = ( 3 3)a, and for r = 4 either a = 0 or a = b, though for r = 4, k = 2 and a = 0 the action is not faithful. It is now the case that gx 0 x 0 = 2b for g either v 2 or some t j. Sometimes, though, w 1 x 0 x 0 will be smaller. For example, this happens with k = 2 and r = 6,8,10, 12, for which cases the first method should be used. In the midst of this chaos, we find that the two methods give the same values of a, b for r = 8, k = 4. Ths could be a case for further investigation. More generally, the initial vector problem should be looked at more closely for those groups that show potential for good coding in other respects, but for now we leave it. A more serious problem is that, even when the initial vector is not in dispute, the greedy decoding algorithm may not work. Here is a particularly bad example, one of many found numerically once we realized that the proof of Property 5 failed. Consider G(6,6,2). Since k = r, we should use the second method to find the initial vector, yielding x 0 = 0.259, Suppose that the received vector is r = 0.667, i so that r x 0 = The correct decoding is g = t 2 v 2 with t 2 v 2 r x 0 = But for every other group element h we have hr x , so the decoding process will never arrive at g = t 2 v 2. Indeed, it will yield I. A relatively minor concern, by comparison, is that relations such as t 2 v 3 = v 3 v 2 t 2 = v 3 t 2 v2 1 mean that Property 7 does not hold for G(r,k,n) with k > 1. Perhaps a different (complex?) initial vector, or a different subgroup sequence, could salvage the situation, but the current decoding scheme fails. A more relevant question is: Are any groups G(r,k,n) with k > 1 suitable for coding purposes? References [1] D. Slepian, Permutation modulation, Proc. IEEE 53 (1965), [2] D. Slepian, Group codes for the Gaussian channel, Bell Syst. Tech. J. 47 (1968),

14 14 HYE JUNG KIM, J. B. NATION AND ANNE SHEPLER [3] I. Ingemarsson, Group codes for the Gaussian channel, in Topics in Coding Theory (Lecture Notes in Control and Information Theory), vol. 128, New York, Springer- Verlag (1989), [4] T. Ericson, Permutation codes, Rapport de Recherche INRIA, no. 2109, Nov [5] T. Mittelholzer and J. Lahtonen, Group codes generated by finite reflection groups, IEEE Trans. on Information Theory, 42 (1996), [6] M. Fossorier, J. Nation and W. Peterson, Reflection group codes and their decoding, preprint Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USA Department of Mathematics, University of North Texas, Denton, TX 76203, USA address: