ON THE NON-ABELIAN GROUP CODE CAPACITY OF MEMORYLESS CHANNELS

. In this work is provided a deﬁnition of group encoding capacity C G of non-Abelian group codes transmitted through symmetric channels. It is shown that this C G is an upper bound of the set of rates of these non-Abelian group codes that allow reliable transmission. Also, is inferred that the C G is a lower bound of the channel capacity. After that, is computed the C G of the group code over the dihedral group transmitted through the 8PSK-AWGN channel then is shown that it equals the channel capacity. It remains an open problem whether there exist non-Abelian group codes of rate arbitrarily close to C G and arbitrarily small error probability.


Introduction
Group codes were introduced by Slepian in [19] as a generalization of binary linear codes. Group codes allow to use non-binary, highly spectral-efficient, geometrically uniform modulations, while inheriting many of the nice structural properties enjoyed by binary linear codes. An overview of the different research lines on group codes developed during these years can be seen at [4,3,5,9,11,17,14,7,13]. One line of research is about the channel-capacity achieved by group codes. Differently from linear and non-linear codes, to speak about ensembles of group codes achieving channel capacity C demands the previous definition of group encoding capacity C G , for each group G over which is based the group code. This fact was studied first by Ahlswede in [1] and the literature cited there. It was shown that C G ≤ C and C G satisfies the converse of the Shannon's encoding theorem, that is, if a group code C has a rate R such that R > C G then its decoding error probability P e (C) is bounded away from zero. Thus, after [1], an ensemble of group codes over a group G transmitted through a channel with capacity C is said to achieve the channel capacity if C G = C and, for any > 0 and any R < C G , there is a code C of the ensemble with rate R such that P e (C) < . (The Shannon's encoding Theorem for C G ).
H. A. Loeliger, in [12], conjectured whether group codes over cyclic groups transmitted trough MPSK-AWGN channels would achieve channel capacity. This conjecture was proved to be right in [4]. To prove this achievement, given a finite Abelian group G, it is defined a G-Symmetric channel in such a way that it is a generalization of MPSK-AWGN channels and the well known symmetric BSC and BEC channels. The formula that defines the group encoding capacity C G is a maxmin choice among suitable weighted capacities of the sub-channels determined by the subgroups of G. Following the random coding exponent of [8] it is derived a formula of random coding exponent for group codes. The randomization is made from a single group code by using the technique proposed in [18]. Finally, the equality C G = C is obtained for G = Z p r transmitted through a p r -PSK-AWGN channel and an example where C G < C is given. On the other hand, the paper [17] uses the asymptotic equipartition property (AEP) to deal with ensembles of group codes, over Abelian groups, with rates achieving the group encoding capacity C G . It is not concerned with the equality C G = C.
In this work we develop most of the above ideas about C G for a special class of non-Abelian groups and show that the group encoding capacity equals the channel capacity. The importance of this equality is clear since any ensemble of group codes with C G < C has not any possibility to achieve the channel capacity. The non-Abelian groups which we will deal are extensions of the form G = Z η p r 1 Z p2 . We show that, also in this case, C G satisfies the converse of the Shannon's encoding theorem, that is, R < C G is a necessary condition for reliable transmissions over G-Symmetric channels. After that, we will show analytically that if G is the dihedral group with eight elements and the channel is 8PSK-AWGN then C G = C. Computer aided proof of this result, for some values of the signal-to-noise ratio E b N0 was presented at [2].
The paper is organized as follows: • In Section 2 is reviewed extension of groups H K and it is shown that the groups (H K) N and H N K N are isomorphic [Proposition 1]. This property is fundamental to define group codes over non-Abelian groups. • In Section 3 is reviewed G-Symmetric channels and the group codes for these channels in accordance to [4]. Then is provided a definition of group encoding capacity C G for non-Abelian groups G = Z η p r 1 Z p2 [Definition 2]. It is shown that this C G is an upper bound for any reliable rate transmission R [Theorem 1]. Also is concluded that R ≤ C G ≤ C, where R is a reliable transmission rate and C is the channel capacity.
It is estimated the C G when G is the dihedral group D 4 and the channel is 8PSK-AWGN. • In Section 4 is shown that the C G for the dihedral group D 4 over the 8PSK-AWGN channel, denoted specifically by C D4 , equals the channel capacity, that is, C D4 = C [Theorem 2]. The proof of this Theorem is divided in four Lemmas on the capacities of the sub-channels that determine the encoding capacity C D4 .
2. Groups and extension of groups 2.1. Notation for groups and related. Cyclic groups of order n will be represented by Z n = {id, a, a 2 , . . . , a n−1 } with a a generator. Sometimes it will be more convenient to write Z n = {0, 1, . . . , n − 1}. This last notation is useful to represent subgroups mZ n = {0, m, . . . , m(n − 1)}. The symbol ⊕ will represent the direct product of groups, that is, H ⊕ K will mean the direct product of the groups H and K. Given a natural number N and a group G, the notation G N will mean the mul- A generalization of the direct product of groups is the extension of groups [16,10]. The extension of groups H by K will be denoted by H K. If G is an Abelian group and H, K are subgroups of G then the symbol + will be used in the sense that H + K = {g ∈ G ; g = h + k, h ∈ H, k ∈ K}. Notice that the group H + K is very different from the group H ⊕ K. The symbol ∼ = will be used for isomorphism of groups, that is H ∼ = K will mean H is isomorphic with K. Finally the notation H ⊂ G will mean "H is a subgroup of G".

2.2.
Direct product of extension of groups: (H K) N . A group G with normal subgroup H G such that the quotient group G/H is isomorphic with a group K is said to be an extension of H by K [16]. Since each element g ∈ G belongs to an unique coset Hk ∈ G/H, g can be written as a "ordered pair" g = hk. The group operation , determines a group isomorphism between G and the extension H by K. The semidirect product and direct product of groups are particular cases of extension of groups. It can be shown that when h k1 2 = h 2 , for some h 2 or else some k 1 , then the extension H by K is a non-Abelian group. In this article, with the purpose of simplicity, the extension will be denoted by H K whereas the direct product will be represented by H ⊕ K and, for any integer N ≥ 1, G N will be the N -fold direct product of G. Proposition 1. If G = H K then: Proof. For the case N = 2, define ϕ : G 2 → (G/H) 2 as ϕ(g 1 , g 2 ) = (g 1 H, g 2 H). Then ϕ((g 11 , g 12 ) * (g 21 , g 22 )) = ϕ(g 11 g 21 , g 12 g 22 ) = (g 11 g 21 H, g 12 g 22 H). On the other hand ϕ(g 11 , g 12 ) * ϕ(g 21 , g 22 ) = (g 11 H, g 12 H) * (g 21 H, g 22 H) = (g 11 g 21 H, g 12 g 22 H), which shows that ϕ is a group homomorphism. Clearly ϕ is surjective with kernel ker(ϕ) = H 2 . Therefore, H 2 is a normal subgroup of G 2 = (H K) 2 and by the First Isomorphism Theorem for groups, Then defining ϕ : G N → (G/H) N as ϕ(g 1 , g 2 ) = (g 1 H N −1 , g 2 H), where g 1 ∈ G N −1 and g 2 ∈ G, we can show, analogously to the case N = 2, that ϕ is a surjective group homomorphism with kernel H N . Therefore [16,10].

G-Symmetric channels, group codes and group encoding capacity
In this Section we will provide the definition of group encoding capacity C G for a particular class of non-Abelian groups. We mention here that, in [4], C G is originally called G-Capacity. We will show that C G ∈ [R, C], where R is a reliable transmission rate and C is the channel capacity. Then, we will apply this estimation of C G for G = D 4 , the dihedral group with eight elements, for transmission over the 8PSK-AWGN channel.
• G acts simple and transitively over the input alphabet X , • G acts isometrically over the output set Y and • p(gy|gx) = p(y|x) for all g ∈ G, for all x ∈ X and for all y ∈ Y.
Some examples of G-Symmetric channels: . The group D 4 has a representation in G(2, R), the set of orthogonal matrices of the plane R 2 , via the With this representation is a straightforward task to verify the three conditions about G-Symmetry on the above Definition 3.1.

Remark 1.
We write here the simply transitive action of D 4 over X 8 which allows to represent the channel by the triple (D 4 , R 2 , p(y|x)) instead of (X 8 , R 2 , p(y|x)). The signals are: In general, the simply transitive action of the group G over the signal constellation X allows to represent the channel as the triple (G, Y, , p(y|x)) instead of (X , Y, p(y|x)). The convenience of the group representation of a G-Symmetric channel comes when we choose a code against the noise of the channel. The natural choice is a group code which is a subgroup of G N . A group code C for a G-Symmetric channel is the image of an injective group homomorphism φ : U → G N , with U an uncoded channel source which must have a group structure such that This implies that the group code C and the uncoded group U must have the following structure; with the associated array of exponents k = k 11 , k 12 , . . . , k 1r k 21 satisfying the conditions: k 11 + k 12 + · · · + k 1r ≤ N η and k 21 ≤ N . For each group code C with array of exponents k consider the arrays of integers l = l 11 , l 12 , . . . , l 1r l 21 such that l ij ≤ j for all i, j. Then, j − l ij ≥ 0 and . With this we construct subgroups U(l) ⊂ U as follows: In particular for l with l ij = j, U(l) = U.
On the other hand if the symbol represents the "addition" of subgroups with k 11 + k 12 ≤ N and k 21 ≤ N . If we consider an array jk ij log(p i ), with k 2j = 0 for j > 1, whereas the encoding rate of the group code C l is R l = log(|U (l)|) l ij k ij log(p i ), with l 2j = 0 for j > 1. By making we see that α ij > 0 and i,j α ij = 1 that allow to consider α ij as a distribution of probabilities on the array k of U. Moreover, α ij is independent of any array l, it depends only on N . Thus the encoding rate R is related to any sub-code rate R l by the equation; , for any array l. Z p2 , we define the Group Encoding capacity C G of the G-symmetric channel (G, Y, p(y|g)) by; with C l the capacity of the G(l)-sub-channel.
Theorem 3.3. Let G and C G be as in Definition 3.2. Let C be a group code with transmission rate R. If R > C G then the error probability of the code is bounded away from zero, that is, there is A > 0 such that P e (C) > A.
Proof. Let C be a group code with rate R such that R > C G . There is a fixed for any array l. On the other hand , for some array l * .
Since R = . Therefore; By the converse of the Shannon's Coding Theorem, the error probability p e (C l * ) is bounded away from zero, i.e., there is A > 0 such that p e (C l * ) > A. By the uniform error property (UEP), p e (C) = p e (C|0) and p e (C l * ) = p e (C l * |0), with 0 ∈ C the N -tuple (id, id, . . . , id), with id the identity element of G. Therefore p e (C) = p e (C|0) ≥ p e (C l * |0) = p e (C l * ) > A.
When the array l is such that l ij = j we have C l = C, which is the capacity of the channel. Then, from Definition 3.2, C G ≤ C. Thus, for reliable group codes, C G can be considered a number in the interval Proof. Since G(l ρ ) = G(l), also C l ρ = C l . On the other hand, The Proposition 2 allows us to simplify the formula of C G to; (3) C G = max α min ρ={1,2,...,r} constellations X (l ijk ), which determine the sub-channels (G(l ijk ), R 2 , p(y|x)), are organized in the Table 1. Table 1. G(l)-Symmetric sub-channels the D 4 -symmetric channel 8PSK-AWGN.
Since the sub-channel (G(l ijk ), R 2 , p(y|x)) is G(l ijk )-Symmetric, its capacity is achieved when the probability distribution over X (l ijk ) is uniform. With this the probability density of the output is For instance, λ l111 (y) = 1 4 (p(y|x 0 ) + p(y|x 1 ) + p(y|x 4 ) + p(y|x 5 )) and the capacity for the respective sub-channel is The densities and capacities of the four sub-channels are shown in the Table 2. Then, the encoding capacity C D4 of the channel is By choosing α 11 = 0, α 12 = 2/3, α 21 = 1/3 and combining the formulas (2) and (4) we have Table 2. Output probability densities λ l ijk and capacities C l ijk of the sub-channels G(l) of the D 4 -symmetric channel 8PSK-AWGN, where p i (y) := p(y|x i ).
The sub-channels generated by the arrays l 110 and l 120 are symmetric over the cyclic groups Z 2 and Z 4 respectively. By Theorem 15 of [4], the expression 3C l110 ≥ 3C l 120 2 ≥ C is reduced to , C}.

The encoding capacity C D4 equals the channel capacity of the 8PSK-AWGN channel
This Section contains the main result of this article: the Theorem 4.3 which states the group encoding capacity equals the channel capacity. The proof of the Theorem will be divided in the following Lemmas 1,2, 3 and 4. Let us begin relating the formula (5) with C D4 = C. We see that the solution is 2C ≤ 3C l111 which in terms of entropies can be written as (6) 3H(λ l111 ) ≥ H(p 0 ) + 2H(λ 121 ).
On the Table 1 it can be seen that the array l 121 generates the whole 8PSK-AWGN channel, the D 4 -Symmetric channel, hence the density λ l121 can be denoted simply as λ := λ l121 . Also, on the same Table 1 it can be seen that the subconstellation {e, b, a 2 , a 2 b} ∼ = Z 2 2 is the input of the sub-channel generated by the array λ l111 . Since the group Z 2 2 , called the "Klein 4-group", is denoted by V in many places of algebraic literature, we denote λ V := λ l111 = 1 4 (p 0 + p 1 + p 4 + p 5 ). With this shortened notations, the inequation (6) becomes:

4.3.
The group encoding capacity C D4 equals the channel capacity C.
Main Theorem. The group encoding capacity C D4 of the 8PSK-AWGN channel equals the Shannon's channel capacity. Proof.

Conclusion
We provided a definition of group encoding capacity C G for group codes over non-Abelian groups transmitted through symmetric channels and showed that C G satisfies the converse of the coding theorem of Shannon, that is, we proved that R < C G is a necessary condition for the Shannon's coding theorem. After, for the particular case of G = D 4 the dihedral group of 8 elements and 8PSK-AWGN channel, we proved that C G achieves the channel capacity. Some still open problems closely related to this work are: • It was shown in [4,17] that the condition R < C G , for Abelian G, is also a sufficient condition to prove the coding theorem of Shannon, that is, it was shown that for each > 0 and each rate R < C G there is a group code C such that P e (C) < . The natural question that arises here is whether R < C G would also be a sufficient condition for the coding theorem if G is non-Abelian. Initial references and guides are [14,15].