Abstract
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, elements, prime, then the group code produced by the encoder is noncontrollable.
1. 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 noiseβAWGN channels [1β4]. In the landmark paper [5], Shannon showed that there exist methods to encode data against channel noise. One key idea behind an error-correcting codeβECCβis the measure of the channel noise in terms of error probability . For binary data, of a given physical channel can be obtained empirically. For instance, if after transmitting ββbits through the channel we observe that βbits were transmitted correctly and one bit was transmitted erroneously, then we can say that the upper bound of , of this channel, is or . 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 , then is the added symbols which are functions of the previous symbols. After that is transmitted, encoded symbols, instead of . The introduction of the additional 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 keyingβPSKβmodulation. 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 . Towards to generalize, Ungerboeckβs technique proposed the matching of a generic group that could represent the output set of an encoder with a discrete set of points from an Euclidean space that could represent the signal constellation from ASK (amplitude shift keying), FSK (frequency shift keying), or PSK modulation, [2, 7β9]. 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 , representing the states of the FSM, inputs group , cyclic group of order prime . Then, our group codes are defined over nonabelian extensions of by . For that, this paper is organized as follows.
In Section 2 is defined the extension of a generic group by the group ; this extension is denoted as . 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 . Finally, is defined the group code , produced by the FSM encoder, as a family of bi-infinite sequences of outputs.
In Section 3, the group code 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 . The main result of this work, which is Theorem 4.7, states that (a) if is nonabelian, with abelian, then the resulting code will have parallel transitions and (b) if both and are nonabelian then the resulting code will be noncontrollable. Therefore, the extension is bad for constructing group codes.
2. Group Extensions and Group Codes
2.1. Group Extensions
Definition 2.1. Given a group with a normal subgroup , consider the quotient group . If there are two groups and such that is isomorphic with and is isomorphic with , then it is said that is an extension of by [11].
We will denote the extension β by β by the symbol , also we will use the standard notations meaning β is isomorphic with β and meaning β normal subgroup of .β When is an extension , each element can be βfactoredβ as an unique ordered pair , and . The semidirect product is a particular case of extension, but also it is known that the semidirect product is a generalization of the direct product . A canonical definition of extension of groups is given in [11, 12]; specially in [12] we find a βpracticalβ way to decompose a given group , with normal subgroup , in an extension . That decomposition depends on the choice of isomorphisms , and a lifting such that , the neutral element of . Then, defining by and , the decomposition with the group operation is isomorphic with , that is, .
Notice that the resulting pair of , of the above operation (2.3), is for some and is the operation on . 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 is the resulting pair of the multiple product , where is some element of . Analogously, for some .
Example 2.2. Consider the direct product group . This abelian group can be decomposed as an extension .
By using the more convenient notation 00 instead of , 010 instead of , and so forth, we have that the normal subgroup is isomorphic with . The quotient group = is isomorphic with . Thus, in an expected way, we have shown that is an extension of .
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 , where is the inputs alphabet, is the alphabet of states of the machine, is the outputs alphabet, is the next-state mapping, and 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 , , and be finite groups. Let and be group homomorphisms defined over an extension such that the mapping defined by
is injective with surjective.
Then, an encoder of a group code is the machine .
The group is called the uncoded information group and is called the encoded information group. To begin working, the encoder needs an initial state and a sequence of inputs , . Then, the encoder will respond with two sequences , , and , in the following way: If we agree that the present time is 0 (zero) and the state represents the present state, then the next integer time is 1 (one) and represents the next state from now. Analogously, the next state from will be and generally will be the next state from . In this way, states with positive indices, , form a sequence of future states.
On the other hand, since is surjective, then must exist at least one pair such that . The state can represent the previous state from the present state . Analogously for , there must exist a pair such that with representing a previous state from and so on is one previous state from . Thus, for a given present state , there are sequences of past states , past outputs , and past inputs such that
Therefore, given bi-infinite sequence of inputs , , and one state , the encoder will response with the sequence , , of outputs while its internal states will have the sequence , . Notice that once made the choice of one initial state , the future relations between the single sequences of inputs and the pair of sequences of outputs and states are bijective, that is, , where is the natural numbers set.
Definition 2.4. A time-invariant group code is the family of bi-infinite sequences produced by the encoder , with . Each sequence is called a codeword [7β9, 15].
Example 2.5. Consider the encoder where defined by and is defined by .
Suppose that the encoder is initialized at state , then for the inputs sequence the encoder states will be and the sequence of encoded outputs will be .
3. 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 is a dynamical system. In this context, the encoder is a realization of [8, 9, 16].
Given a codeword and a set of consecutive indices , the projection of the codeword over these indices will be . Analogously, , , and so on. With this notation, the concatenation of two codewords , in the instant is a sequence defined by
Definition 3.1. If is an integer greater than one, then a group code is said -controllable if, for any pair of codewords and , there are a codeword and one integer such that the concatenation is a codeword of the group code [7, 10, 15].
It is said that a natural number is the index of controllability of a group code when . 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 to be -controllable is a sufficient condition for to be controllable.
3.1. Trellis of a Group Code
The triplets of the set , where is defined by (2.4), can be represented graphically. In the context of graph theory [17], they are called edges whose vertexes set is and the graph is called state diagram labeled by . In Figure 1, the full state diagram of the code generated by the FSM , 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 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 and , the transitions and are said concatenated when . Hence, a bi-infinite trellis path of transitions is a sequence such that and are concatenated for each . The set of trellis paths form the trellis diagram. Since each codeword passes only by one state at each unit of time, the relation between the codewords and paths is bijective. Again from Example 2.5, consider the inputs sequence , such that , , , , , , and for all . The response path is such that , , , , , , and for all . This response path is shown by a traced line in Figure 1.
Definition 3.2. Two states and are said to be connected when there are a path and indices such that with and such that and .
Theorem 3.3. Let be a group code produced by the encoder . If there are two states and for which there is no a finite path of transitions connecting them, then is noncontrollable.
Proof. On contrary, there is such that is the controllability index of . Let be one codeword passing by the state at time ; let be a codeword passing by the state at time , . There must exist with its respective path such that and and , a finite path, connecting and , a contradiction.
Equivalently, we can say that two states and are connected when there is a finite sequence of inputs such that
Theorem 3.4. Given an encoder , consider the family of state subsets , recursively defined by then (1)each is a subgroup of , (2) is normal in , for all , (3)if , then , (4)if the group code is controllable, then for some .
Proof. (1) Consider . Since is surjective, there exist and with and such that and . Hence, , and thus .
(2) Clearly, . For , suppose that , for all . Given and , consider , where , , . Hence, , because .
(3) Given , there are and such that . Since , . Hence, .
(4) If not, then there is such that for any . Then, the neutral state and are not connected by any finite trellis path. Therefore, the group code is noncontrollable.
In Figure 1ββ, , , therefore the code is controllable.
Definition 3.5. Two different transitions and are parallels if and and .
The Hamming distance between two codewords and is defined as the number of components which are different. The minimal distance of a group code is One desirable property of a group code is a high ; the greater , the better the capability to correction of errors. Clearly, when the trellis of a group code has parallel transitions, then and therefore a group code with parallel transitions will not be a good group code.
Lemma 3.6. Consider an encoder . Let and be subsets of the trellis section group such that , the transitions outcoming from the neutral state , and , the transitions incoming into the neutral state . Also, let and be subsets of such that and , then (1) and , (2)both and are normal subgroups of , (3)if , then has parallel transitions, (4)if is nonabelian and the states group is abelian, then has parallel transitions.
Proof. (1) We have and , where is defined in (2.4).
(2) Immediate.
(3) There exists , with such that . Since of (2.4) is injective, . Therefore, the transitions and are parallels.
(4) The states group being abelian implies that are abelian factor groups. Then, the commutators subgroup is a subgroup of . But is nonabelian, then . Therefore, from the above item (2.3), has parallel transitions.
4. Nonabelian Group Codes with as information Group Are Not Good
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 .
Definition 4.1. Given a finite group and a subgroup , the index of in , denoted by , is the number of different cosets of in .
If denote the orders of and , respectively, then . It is possible to represent graphically this index such as in Figure 2 where is represented the index . We see that if , then , and if then .
| (a) If |
| (b) If |
Definition 4.2. Given a group , the group of commutators of is the subgroup .
Lemma 4.3. Let be an extension which is a -group. If , then , and , for each .
Proof. Since is a group homomorphism, the image is in the commutators subgroup of . If , then the lemma holds trivially (Figure 2(a)). If , then by the long commutators theorem from [18], there are and such that . Now, consider and such that . We have and . Therefore, (Figure 2(b)).
Again, since is a group homomorphism, is in the commutators subgroup of . Then, with very similar arguments, we can proof that if , then and . Continuing in the same way, we conclude that and , for any .
Lemma 4.4. Let be an extension which is a -group. Consider the subgroups defined in (3.3). Then, for each , either each is abelian or .
Proof. Since is cyclic and has at most order , we have that both and are abelian. Then, let be such that are all abelian with nonabelian. Then, there are such that . Also, there must be and , with , such that and . Then, From this, and . Since the order of is , we have that . By Lemma 4.3, and , for each . Therefore, either is abelian or .
Suppose now that we do not have the information about the order of ; that is, we cannot use the hypothesis that is a -group. In this case, we need to consider as a generic and finite group. By looking back, again, the family defined by (3.3), we will show that when , each must be a -group. In that direction, we need to consider the subgroup of Lemma 3.6 and from this the second projection of the kernel of : Clearly, is a normal subgroup of and it is isomorphic to and we have the following lemma.
Lemma 4.5. Consider the encoder . Also, consider the subgroup defined in (4.2), then (1)if there are and , then , for , (2)if , then , for .
Proof. (1) Since , then .
(2) Given such that , suppose there is some such that . For the subgroup , we have that is a coset where each element is , for some . Hence with . But, since , there is at least one such that , in contradiction with .
Theorem 4.6. Consider the encoder , where is prime. Then, each of (3.3) must be a -group.
Proof. By induction over , for we have or . Now, suppose that there is a natural number such that , for all . We have that the subgroup has elements and each of its elements has order , . If , then , where each is a prime and . There must be an element such that .
Let and be such that , then . Hence, .
If , then , because . By Lemma 4.5, and , a contradiction.
If , then , a contradiction.
Theorem 4.7. Consider the encoder , where is nonabelian and is a positive prime, then (1)if is abelian, then the group code has bad minimal Hamming distance; (2)if 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 .
(2) If is not a -group, then by Theorem 4.6 the resulting code is noncontrollable. If is a -group, then is also a -group, then by Lemma 4.4ββ is abelian, a contradiction.
5. Conclusion
We have shown that there are no controllable group codes defined on the nonabelian extension , with prime and for any finite nonabelian group . When is abelian, the group code will have parallel transitions and therefore distance limitations. In contrast, for the cases and , 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 , where is the symmetries of the square and the symbol denotes the semidirect product. This provides a strong clue about the controllability of some -groups with information groups or .