On the Uncontrollability of Nonabelian Group Codes with Uncoded Group Z p

Error-correcting encoding is a mathematical manipulation of the information against transmission errors over noisy communications channels. One class of error-correcting codes is the so-called group codes. Presently, there are many good binary group codes which are abelian. A group code is a family of bi-infinite sequences produced by a finite state machine FSM homomorphic encoder defined on the extension of two finite groups. As a set of sequences, a group code is a dynamical system and it is known that well-behaved dynamical systems must be necessarily controllable. Thus, a good group code must be controllable. In this paper, we work with group codes defined over nonabelian groups. This necessity on the encoder is because it has been shown that the capacity of an additive white Gaussian noise AWGN channel using abelian group codes is upper bounded by the capacity of the same channel using phase shift keying PSK modulation eventually with different energies per symbol. We will show that when the trellis section group is nonabelian and the input group of the encoder is a cyclic group with, p elements, p prime, then the group code produced by the encoder is noncontrollable.


Introduction
Data to be transmitted through a noisy channel may suffer impairments when it arrives to its destination.The most known channel noise is the Gaussian noise, which is modeled as a random signal having a normal probabilistic distribution.The channels suffering Gaussian noise are called additive white Gaussian noise-AWGN channels 1-4 .In the landmark paper 5 , Shannon showed that there exist methods to encode data against channel noise.One key idea behind an error-correcting code-ECC-is the measure of the channel noise in terms of error probability P e .For binary data, P e of a given physical channel can be obtained empirically.For instance, if after transmitting 10 15 bits through the channel we observe that 10 15 − 1 bits were transmitted correctly and one bit was transmitted erroneously, then we can say that the upper bound of P e , of this channel, is 10 −15 or P e < 10 −15 .This error probability can be reduced by enhancing the hardware components of the channel or alternatively by using ECC.Mostly the use of ECC on the data to be transmitted is more economic than the enhancing of hardware components of the physical channel.That is why an ECC is important.Classically, the essence of an ECC is the splitting of the information data to be transmitted in packages with constant size, let us say m, then is the added k symbols which are functions of the previous m symbols.After that is transmitted, n m k encoded symbols, instead of m.The introduction of the additional k symbols reduces the data rate through the channel.To maintain the original velocity of the data transmission, the physical channel demands either more power or more bandwidth.
In 1981, Ungerboeck in 6 introduced a way to encode data against transmission errors, without increasing the consumption of bandwidth by using a technique called set partitioning map.This technique matches the output symbols of a classical binary convolutional encoder and the signal constellation from a phase shift keying-PSKmodulation. Mathematically, the output symbols of a binary convolutional encoder with the modulo-2 addition constitute an abelian group, whereas the PSK constellation constitutes a discrete set of points from a bidimensional vector space isomorphic with the plane R 2 .Towards to generalize, Ungerboeck's technique proposed the matching of a generic group Y that could represent the output set of an encoder with a discrete set of points from an Euclidean space R n that could represent the signal constellation from ASK amplitude shift keying , FSK frequency shift keying , or PSK modulation, 2, 7-9 .In this direction in 7 is introduced the wide sense homomorphic encoder which is a finite state machine FSM defined over extension of groups.The bi-infinite outputs of this encoder are the codewords and constitute the group code.But, one important result that motivates our work was given in 8 where it was shown that any AWGN channel using group codes over abelian groups has its capacity upper bounded by some uncoded AWGN channel capacity using PSK modulation.Thus, nonabelian and well-behaved group codes could surmount this PSK limit.In the sense of control theory, a group code is well behaved when it is controllable and observable.Classical group codes over binary groups always are well behaved, that is why there is not any concern about control consideration of these kind of codes.This well behavior also is true for some especial cases of abelian group codes.In the current paper, we deal with nonabelian group codes but we do not construct any new code.Instead of it, we study the class of group codes produced by an FSM encoder with a group S, representing the states of the FSM, inputs group Z p {0, 1, 2, . . ., p − 1}, cyclic group of order prime p.Then, our group codes are defined over nonabelian extensions of Z p by S. For that, this paper is organized as follows.
In Section 2 is defined the extension of a generic group U by the group S; this extension is denoted as U S. Then is defined the FSM encoder of a group code which also is called ISO input/state/output machine.The next state mapping and the encoder output mapping are defined over the extension U S. Finally, is defined the group code C, produced by the FSM encoder, as a family of bi-infinite sequences of outputs.
In Section 3, the group code C is presented as a dynamical system in the sense of 10 .Also, a graphical description of a group code known as a trellis is presented.It is established that the trellis diagram is a set of paths of transitions between states.After given the control definition, a sufficient condition of noncontrollability is made in Theorem 3.4.Also are presented the definition and conditions about parallel transitions that are used directly in our main result.
In Section 4, we present our original contributions about the noncontrollability of group codes produced by encoders defined on nonabelian extensions Z p S. The main result of this work, which is Theorem 4.7, states that a if Z p S is nonabelian, with S abelian, then the resulting code will have parallel transitions and b if both Z p S and S are nonabelian then the resulting code will be noncontrollable.Therefore, the extension Z p S is bad for constructing group codes.

Group Extensions
Definition 2.1.Given a group G with a normal subgroup N, consider the quotient group G/N.If there are two groups U and S such that U is isomorphic with N and S is isomorphic with G/N, then it is said that G is an extension of U by S 11 .
We will denote the extension "U by S" by the symbol U S, also we will use the standard notations U ∼ N meaning "U is isomorphic with S" and N G meaning "N normal subgroup of G." When G is an extension U S, each element g ∈ G can be "factored" as an unique ordered pair u, s , u ∈ U and s ∈ S. The semidirect product U S is a particular case of extension, but also it is known that the semidirect product is a generalization of the direct product U × S. A canonical definition of extension of groups is given in 11, 12 ; specially in 12 we find a "practical" way to decompose a given group G, with normal subgroup N, in an extension U S. That decomposition depends on the choice of isomorphisms υ : the decomposition U S with the group operation is isomorphic with G, that is, g u, s .Notice that the resulting pair of u 1 , s 1 * u 2 , s 2 , of the above operation 2.3 , is u , s 1 s 2 for some u ∈ U and s 1 s 2 is the operation on S.This property allows us to do not be concerned to obtain an explicit result when multiple factors are acting.For instance, in the proof of some Lemmas, it will be enough to say that u , s This abelian group can be decomposed as an extension Z 2 Z 2 2 .
By using the more convenient notation 00 instead of 0, 0 , 010 instead of 0, 1, 0 , and so forth, we have that the normal subgroup N {000, 100} Z 3 2 is isomorphic with Z 2 .The Mathematical Problems in Engineering Thus, in an expected way, we have shown that Z 3 2 is an extension of Z 2 Z 2 2 .

Finite State Machines and Group Codes
Finite state machines FSM are a subject of automata theory.Arbib in 13 describes a FSM as a quintuple M I, S, O, δ, ξ , where I is the inputs alphabet, S is the alphabet of states of the machine, O is the outputs alphabet, δ : I × S → S is the next-state mapping, and ξ : I × S → O is the output mapping.Following 8, 9, 14 and by making modifications on the FSM notation, suitable for our context of group codes, it is given the definition of an encoder as follows.
Definition 2.3.Let U, S, and Y be finite groups.Let ν : U S → S and ω : U S → Y be group homomorphisms defined over an extension U S such that the mapping Ψ : U S → S×Y ×S defined by is injective with ν surjective.
Then, an encoder of a group code is the machine M U, S, Y, ν, ω .
The group U is called the uncoded information group and Y is called the encoded information group.To begin working, the encoder needs an initial state s 0 ∈ S and a sequence of inputs {u i } n i 1 , u i ∈ U.Then, the encoder will respond with two sequences {s i } n i 1 , s i ∈ S, and {y i } n i 1 , y i ∈ Y in the following way:

2.5
If we agree that the present time is 0 zero and the state s 0 represents the present state, then the next integer time is 1 one and s 1 represents the next state from now.Analogously, the next state from s 1 will be s 2 and generally s i will be the next state from s i−1 .In this way, states with positive indices, {s i } n i 1 , form a sequence of future states.On the other hand, since ν is surjective, then must exist at least one pair u 0 , s −1 such that s 0 ν u 0 , s −1 .The state s −1 can represent the previous state from the present state s 0 .Analogously for s −1 , there must exist a pair u −1 , s −2 such that ν u −1 , s −2 s −1 with s −2 representing a previous state from s −1 and so on s −i is one previous state from s {−i 1} .Thus, for a given present state s 0 , there are sequences of past states {s i } −1 i −n , past outputs {y i } −1 i −n , and past inputs {u i } 0

2.6
Therefore, given bi-infinite sequence of inputs {u i } i∈Z , u i ∈ U, and one state s 0 ∈ S, the encoder M U, S, Y, ν, ω will response with the sequence {y i } i∈Z , y i ∈ Y , of outputs while its internal states will have the sequence {s i } i∈Z , s i ∈ S. Notice that once made the choice of one initial state s 0 , the future relations between the single sequences of inputs and the pair of sequences of outputs and states are bijective, that is, Suppose that the encoder is initialized at state s 0 00, then for the inputs sequence {1, 1, 0, 0, 1, 1, 0, 1, 0, 1} the encoder states will be {10, 11, 11, 11, 01, 00, 00, 10, 01, 00} and the sequence of encoded outputs will be {010, 011, 101, 101, 111, 110, 000, 010, 001, 110}.

Control and Group Codes
Each codeword of a group code satisfies the definition of a trajectory of a dynamical system in the sense of Polderman and Willems 10 .From this, each group code C is a dynamical system.In this context, the encoder M U, S, Y, ν, ω is a realization of C 8, 9, 16 .Given a codeword y and a set of consecutive indices {i, i 1, . . ., j − 1, j} i, j , the projection of the codeword over these indices will be y| i,j {y i , y i 1 , . . ., y j }.Analogously, y| i,j {y i , y i 1 , . . ., y j−1 }, y| i, ∞ {y i , y i 1 , . ..}, and so on.With this notation, the concatenation of two codewords y 1 , y 2 ∈ C in the instant j is a sequence y 1 ∧ j y 2 defined by

3.1
Definition 3.1.If L is an integer greater than one, then a group code C is said L-controllable if, for any pair of codewords y 1 and y 2 , there are a codeword y 3 and one integer k such that the concatenation y 1 ∧ k y 3 ∧ k L y 2 is a codeword of the group code C 7, 10, 15 .

Mathematical Problems in Engineering
It is said that a natural number l > 1 is the index of controllability of a group code C when l min{L; C is L-controllable}.Any applicable group code, for correction of errors of transmission and storage of information, needs to have an index of controllability.Shortly, when a code has an index of controllability, then it is said that it is controllable 10 .Clearly, a code C to be L-controllable is a sufficient condition for C to be controllable.

Trellis of a Group Code
The triplets s, ω u, s , ν u, s of the set {Ψ u, s } u,s ∈U S , where Ψ is defined by 2.4 , can be represented graphically.In the context of graph theory 17 , they are called edges whose vertexes set is S and the graph is called state diagram labeled by ω u, s .In Figure 1, the full state diagram of the code generated by the FSM M Z 2 , Z 2 2 , Z 3 2 , ν, ω , from Example 2.5, is shown between the times 2 and 3; also it is repeated between the times 3 and 4 and so on until times 4 and 5.In the context of coding theory the elements of {Ψ u, s } u,s ∈U S are called transitions or branches.The expansion in time of the state diagram is called trellis diagram.This is made by concatenating at each time unit separate state diagram.For two consecutive time units i and i 1, the transitions b i s i , ω u i 1 , s i , ν u i 1 , s i and b i 1 s i 1 , ω u i 2 , s i 1 , ν u i 2 , s i 1 are said concatenated when s i 1 ν u i 1 , s i .Hence, a biinfinite trellis path of transitions is a sequence b {b i } i∈Z such that b i and b i 1 are concatenated for each i ∈ Z.The set of trellis paths form the trellis diagram.Since each codeword y passes only by one state s at each unit of time, the relation between the codewords y and paths b is bijective.Again from Example 2.5, consider the inputs sequence {u i } i∈Z , such that u 1 1, u 2 1, u 3 0, u 4 0, u 5 1, u 6  Definition 3.2.Two states s and r are said to be connected when there are a path b and indices i, j ∈ Z such that b| i,j {b i , b i 1 , . . ., b j } with b i s i , ω u i 1 , s i , ν u i 1 , s i and b j s j , ω u j 1 , s j , ν u j 1 , s j such that s s i and r ν u j 1 , s j .
Theorem 3.3.Let C be a group code produced by the encoder M U, S, Y, ν, ω .If there are two states s ∈ S and r ∈ S for which there is no a finite path of transitions connecting them, then C is noncontrollable.
Proof.On contrary, there is l > 1 such that l is the controllability index of C. Let y 1 be one codeword passing by the state s at time k; let y 2 be a codeword passing by the state r at time k L, L ≥ l.There must exist y 3 ∈ C with its respective path b 3 such that y a finite path, connecting s and r, a contradiction.Equivalently, we can say that two states s and r are connected when there is a finite sequence of inputs {u i } n i 1 such that Theorem 3.4.Given an encoder U, S, Y, ν, ω , consider the family of state subsets {S i }, recursively defined by 4 if the group code is controllable, then S S k for some k.
Proof. 1 Consider r, s ∈ S i .Since ν is surjective, there exist u 1 , s 1 and u 2 , s 2 with s 1 , s 2 ∈ S i−1 and u 1 , u 2 ∈ U such that r ν u 1 , s 1 and s ν u 2 , s 2 .Hence, sr ν u 1 , s 1 * ν u 2 , s 2 ν u 3 , s 1 s 2 , u 3 ∈ U and thus sr ∈ S i . 2 Clearly, S 0 S 1 .For i > 1, suppose that S j−1 S j , for all j ≤ i.Given s ∈ S i 1 and Then, the neutral state e ∈ S k ⊂ S and s are not connected by any finite trellis path.Therefore, the group code is noncontrollable.
In Figure 1 2 where is represented the index Z p S 1 : Z p S 0 .We see that if S 1 S 0 , then Z p S 1 : Z p S 0 1, and if S 1 / S 0 then Z p S 1 : Z p S 0 p.Proof.Since S 1 is cyclic and S 2 has at most order p 2 , we have that both S 1 and S 2 are abelian.Then, let i ≥ 2 be such that S 1 , S 2 , . . ., S i are all abelian with S i 1 nonabelian.Then, there are s 1 , s 2 ∈ S i 1 such that s 1 s 2 / s 2 s 1 .Also, there must be u 1 , u 2 ∈ Z p and r 1 , r 2 ∈ S i , with r 1 r 2 r 2 r 1 , such that s 1 ν u 1 , r 1 and s 2 ν u 2 , r 2 .Then,

Conclusion
We have shown that there are no controllable group codes defined on the nonabelian extension Z p S, with p prime and for any finite nonabelian group S. When S is abelian, the group code will have parallel transitions and therefore distance limitations.In contrast, for the cases Z 2 2 S and Z 2 2 S, there are important examples of controllable and nonabelian group codes such as the Wei code 19 which has a trellis section isomorphic to the nonabelian 2-group Z 2 2 D 8 , where D 8 is the symmetries of the square and the symbol denotes the semidirect product.This provides a strong clue about the controllability of some p-groups with information groups Z p n or Z p n .
where N {1, 2, 3, . ..} is the natural numbers set.Definition 2.4.A time-invariant group code C is the family of bi-infinite sequences y {y i } i∈Z produced by the encoder M U, S, Y, ν, ω , with y i ω u i , s i−1 .Each sequence y {y i } i∈Z is called a codeword 7-9, 15 .Example 2.5.Consider the encoder M

Nonabelian Group Codes with Z p as information Group Are Not Good
Two different transitions s 1 , ω u 1 , s 1 , ν u 1 , s 1 and s 2 , ω u 2 , s 2 , ν u 2 , s 2 are parallels if s 1 s 2 and ν u 1 , s 1 ν u 2 , s 2 and ω u 1 , s 1 / ω u 2 , s 2 .The Hamming distance d H y 1 , y 2 between two codewords y 1 {y 1 i } i∈Z and y 2 {y 2 i } i∈Z is defined as the number of components which are different.The minimal distance of a group code is d min min{d H y 1 , y 2 ; y 1 , y 2 are different codewords}.3.4 One desirable property of a group code is a high d min ; the greater d min , the better the capability to correction of errors.Clearly, when the trellis of a group code has parallel transitions, then d min 1 and therefore a group code with parallel transitions will not be a good group code.Consider an encoder U, S, Y, ν, ω .Let B and B − be subsets of the trellis section group {Ψ u, s } u,s ∈U S such that B { e, ω u, e , ν u, e ; u ∈ U}, the transitions outcoming from the neutral state {e}, and B − { s, ω u, s , ν u, s ; ν u, s e}, the transitions incoming into the neutral state {e}.Also, let H and H − be subsets of U S such that H U {e} { u, e ; u ∈ U} and H − Ker ν { u, s ; ν u, s e}, then 1 H ∼ B and H − ∼ B − , 2 both H and H − are normal subgroups of U S, 3 if H ∩ H − / { e, e }, then {Ψ u, s } u,s ∈U S has parallel transitions, 4 if U S is nonabelian and the states group S is abelian, then {Ψ u, s } u,s ∈U S has parallel transitions.The states group S being abelian implies that G/H ∼ G/H − are abelian factor groups.Then, the commutators subgroup U S is a subgroup of H ∩ H − .But U S is nonabelian, then U S / { e, e }.Therefore, from the above item 2.3 , {Ψ u, s } u,s ∈U S has parallel transitions.By using group extension of groups, subgroup of commutators, Theorem 3.4, and other group theory ideas, we show here that we must search for good nonabelian group codes outside the extension Z p S. Given a finite group G and a subgroup H ⊂ G, the index of H in G, denoted by G : H , is the number of different cosets of H in G.If |G|, |H| denote the orders of G and H, respectively, then G : H |G|/|H|.It is possible to represent graphically this index such as in Figure 4.

2 :
The intersection Z p S 1 ∩ Z p S when Z p S 0 ⊂ Z p S .Since ν is a group homomorphism, the image ν Z p S 0 S 1 is in the commutators subgroup S of S. If S 1 S 0 , then the lemma holds trivially Figure2a .If S 1 / S 0 , then by the long commutators theorem from 18 , there are s ∈ S 1 − S 0 and a 1 , a 2 , ..., a t ∈ S ∈ Z p S and u, s / ∈ Z p S 0 .Therefore, Z p S 1 ⊂ Z p S Figure2b .Again, since ν is a group homomorphism, ν Z p S 1 S 2 is in the commutators subgroup S of S.Then, with very similar arguments, we can proof that if S 2 / S 1 , then Z p S 2 ⊂ Z p S and ν Z p S 2 S 3 ⊂ S .Continuing in the same way, we conclude that Z p S and S i ⊂ S , for any i ≥ 1.Let Z p S be an extension which is a p-group.Consider the subgroups {S i } defined in 3.3 .Then, for each i, either each S i is abelian or S i ⊂ S .
Definition 4.2.Given a group G, the group of commutators of G is the subgroup G {aba −1 b −1 ; a, b ∈ G}.Lemma 4.3.Let Z p S be an extension which is a p-group.If Z p S 0 ⊂ Z p S , then Z p S i ⊂ Z p S ,and S i ⊂ S , for each i ≥ 1. Proof.t .Now, consider u ∈ Z p and {u 1 , u 2 , . . ., u t } ⊂ Z p such that u, s u 1 , a 1 * u 2 , a 2 * • • • * u t , a t * u 1 , a 1 −1 * u 2 , a 2 −1 * • • • * u t , a t −1 .We have u, s 1 s 2 / s 2 s 1 , ν u 1 , r 1 * ν u 2 , r 2 / ν u 2 , r 2 * ν u 1 , r 1 , ν u 1 , r 1 * u 2 , r 2 * u 1 , r 1 −1 * u 2 , r 2 −1 / e,1 if S is abelian, then the group code has bad minimal Hamming distance; 2 if S is nonabelian, then the group code is noncontrollable.Proof. 1 By Lemma 3.6, the group code has parallel transitions and by 3.4 has d min 1. 2 If S is not a p-group, then by Theorem 4.6 the resulting code is noncontrollable.If S is a p-group, then Z p S is also a p-group, then by Lemma 4.4 S is abelian, a contradiction.