On the Matrix and Additive Communication Channels

The notion of a communication channel is one of the key notions in information theory but like the notion “information” it has not any general mathematical definition. The existing examples of the communication channels: the Gaussian ones; the binary symmetric ones; the ones with symbol drop-out and drop-in; the ones with error packets etc., characterize the distortions which take place in information conducted through the corresponding channel.


Introduction
We confine our discussion to the following situation.Let { } 0,1 B = be a binary alphabet and B * be the set of all words with finite length in the alphabet, B .
We take the dictionary function as the following partial mapping: Saying a communication channel, we mean an arbitrary multi-valued mapping, having the following form: where is some dictionary function.As to the content, equality (1) means that when the word х is transferred we have one of the words ( ) i x Ψ at the exit.
Below we take ( ) = without any loss of generality.
We denote the set of all binary words with the length n by n B ; below the terms, "a word" and "a Boolean vector" are synonyms.
Example. 1) Mapping ( 1) is called a standard communication channel if it has a limited number of distortions of the form: 0 1 → , 1 0 → , where ( ) Besides, we say that there are no more than t errors in the channel if k v does not exceed t .(Here k v is Hamming weight of the word k v ).On the other hand, the following holds: where is the cardinality of the sphere with the radius t in n B .
The notion of the code that corrects the errors of the channel Ψ is completely analogous to the classic defi- nition of the code, correcting the distortions of the form: 0 1 → , 1 0 → .Definition 1.The code  corrects the errors of the channel: , if the following is valid: for i , 0, j N = ; p , 0, q m = .Condition (2) means that consequences of errors are different; hence we can restore one to one the initial information at the exit.The decision process at the exit usually is formalized in the form of the "decoding table" [1]: Error "correction" through the table takes place as follows.According to definition, every "transferred" word x is transformed by the channel Ψ into ( ) , which is at least in one of the columns of the table.Then the code vector in the first row of any row is the "prototype" of the transferred word.
It is clear that if the word y belongs to the only one of the columns in the table, then the "decoding" process leads to a right result.
Condition (2) can be formulated in a little different way using the notion of "neighborhood" which gives certain advantage when making estimates of the cardinality of the correcting code.
The neighborhood of the k th order of the word n x B ∈ built up with respect to the set: Is formed by the following induction: Condition (3) shows that the neighborhood of the k th order of the word n x B ∈ is the union of the neighborhood of the 1st order of all words belonging to the neighborhood of the ( ) In the term of the neighborhood condition (2) of error correction takes the following form: , if We denote by ( ) V Ψ the code correcting the errors of the channel Ψ .In the terms of the above introduced notions for the given channel Ψ the problem is to build the code of the maximum cardinality ( ) V Ψ .It is obvious that this cardinality depends on the "structure" of Ψ .
Among the codes ( ) V Ψ the so called perfect codes are of special interest.Definition 2. The code ( ) Assume that for all We consider the graph ( ) no less than three it is necessary and sufficient that V be an error correcting code of the algebraic channel Ψ .
Further we discuss a special but having certain interest type of communication channel which is carried out by linear mappings, ( )

Matrix Channels [3]
Let { } 2 0,1 F = be a finite field of two elements and , p g M be the set of matrices of the order ( ) p g × with the elements belonging to the field 2 F with the usual operations of addition and multiplication for p g then the set M , defines a matrix channel in the sense of (1):

Examples.
2) Let such "errors" take place in a "real" channel, which are connected with wrong reading of adjacent letters of the transferred vector, ( ) ; and this means the following transformation: This situation can be modelized by the matrix channel  through the channel we have a vector of the following form at the exit: ( ) 3) if a "drop-out" of symbols takes place in the channel, i.e. the length of the word is changed, then it can be presented in the matrix form as follows.Let ( )  is the initial word in which just one symbol can be lost.We discuss the following set of matrices belonging to 1, n n M − : The notion of the code that corrects the errors of the matrix channel M is completely analogous to the classic definition of the code, correcting the distortions of the form: 0 1 if the following condition is valid: ; and p , 0, q m = .
The neighborhood of the k th order of the word In the terms of neighborhood the error correction condition becomes as follows: , if

The Group Matrix Channels
Let ( ) GL n be the group of the non-degenerated matrices of the order n on the field 2 F and G be the subgroup of ( )

2
GL n .We discuss the matrix channel generated by the subgroup: where m is a divisor of the number: Moreover, the transitive set: coincides with the neighborhood of the first order of the point These neighborhoods do not intersect and thus, form the partition n B .Consequently, if we take an arbitrary representative from each transitive set, we will have a code, correcting the errors of the group channel, G .
Lemma 1.For the group matrix channel G , any code containing one representative of all transitive sets, is a code with the maximum cardinality, correcting the errors of the channel G .
Proof.As it was mentioned, the code V , built as it was said above, corrects the errors of the matrix channel, G .On the other hand, if some code V correcting the errors of the group channel G contains more points that the number of the transitive sets then at least one of these transitive sets contains two points of V which contradicts condition (4).Q. E. D.
The above Lemma completely describes all the codes of the maximum cardinality, correcting the errors of the group channel G .
The cardinality of the neighborhood ( ) In other words, the following formula is valid: ( ) The cardinality of the code ( ) V G can be expressed by Burnside's Lemma [4] [5].Let ( ) i N M be the set of the motionless points of the transformation i M or in another way (which is the same) let it be the set of the eigen vectors of the matrix i M corresponding the eigen value 1 λ = , that is, let it be the set of the solutions of the following equation: , Lemma (Burnside's) 2. The following formula holds true: ( ) ( ) Examples.4) Let T be the transformation of the cyclic shift in n B : ( ) ( ) and T M be the matrix, corresponding to this transformation: We discuss the group matrix channel According to the definition, this channel operates as follows.If the word n x B ∈ is put in then we get one of the cyclic shifts of this word at the exit.We call the positive integer d the period of the word x if d is the smallest integer for which Then the neighborhood of the first order of the word x has the following form: It is clear that the first order neighborhoods carry out a partitioning of n B into classes of equivalence .If d N is the number of the equivalence classes the elements of which have the period d then the following ob- viously holds: Let us note that the maximum cardinality code ( ) V G is any set of the representatives of the transitive sets and its cardinality is given by Formula (5) which has the following form for this case: Through the standard calculation technique, we get from ( 6) and ( 7) the well-known expression: ( ) ( ) ( ) where ( ) q ϕ is Euler's function which gives the amount of the numbers less than q and which are coprime with respect to it.
In particular, if n p = is a prime number, then: 5) Let there be a communication channel through which the transmitted word: ( ) is transformed into the binary word: ( ) where either 2 1 Let us describe the physical meaning of this channel.Saying "transmittance of the word x through the communication channel" we understand successive transmittance of symbols or, as they say, transmittance of the pulses (signals) , can have distortions defined by the directly preceding symbols.
Thus, having the symbol 2i х at the exit, we can get either 2i х or 2 1 2 i i х х − ⊕ .Now we give this description of the channel by the matrix "language".Let we have the set of the matrices where 0 M is the unit matrix: We discuss the group matrix channel G the constituent of which is the set М .As any matrix of М coin- cides with its inverse matrix in the group G , i.e.
( ) and G is consisted of all possible products of the matrices of M ; therefore, the order of the group G is 2 n .It follows from the description of the channel that the code, correcting its errors, also corrects the errors of the channel with overlay.The converse proposition also is true.
Let us partition the group G into the non-intersecting sets i Q , and i Q be the set of the matrices generated the products of any i different elements, belonging to M\{M 0 }, i.e. the matrix Talking figuratively if we enumerate the matrix rows of the group G from the top to the bottom, then the set is consisted of all matrices having the dimension 2 2 n n × and which have two units in their i -th rows with odd numbers on their diagonal positions and immediately on the right, but in the rows numbered by 2n i − the unit is only in a diagonal position.The other elements of the matrices are zero.

It immediately follows from the definition of the set
and for any matrix , for , and , then it follows from Lemma 2 that for the maximum cardinality code ( ) 2n V G B ⊆ the following holds true: Let us discuss Example 5 for 2 n = , i.e. that there is a word 1 2 3 4 x x x x x B = ∈ which can be transformed into one of the words: when transmitted through the channel.For the given case the set of the matrices is the following: The group channel having the set M as its constituent is consisted of the following matrices: Let us find the set of the motionless points of the transformation for each element of the group G .As it was said above the set of the solutions of the equation: corresponds to the set For the matrix 0 G Equation ( 8) is as follows: ( ) ( ) and the set of solutions of it is the set 4 B .Consequently, ( ) Then, from (8) for the cases: 0 0 0 0 , 0 0 0 1 , 0 0 1 0 , 0 0 1 1 , 0 1 0 0 , 0 1 0 1 , 0 1 1 0 , 0 1 1 1 , 0 0 0 0 , 0 0 0 1 , 0 0 1 0 , 0 0 1 1 , 1 0 0 0 , 1 0 0 1 , 1 0 1 0 , 1 0 1 1 , 0 0 0 0 , 0 0 0 1 , 0 1 0 0 , 0 1 0 1 .
And, applying Lemma 2 we get the cardinality of the code ( ) ) 4 Let us discuss a little modified channel of Example 2. Namely, we take that when transmitting the vector ( ) x x x x =  some "transposition" errors of the following form take place: taking into account that such inversions can take place in a few places.
In the terms of matrix channels the model is as follows.We have the set of the matrices , where 0 M is the unit matrix: Considerations analogous in the preceding example let us establish the following facts.The matrix channel G consisting of all possible products of the elements the of the group G is 2 n and the code, correcting the errors of the channel G also corrects the errors of the channel with the transpositions.Then, following the same logic and, using Formula (5) for the maximum cardinality code ( ) 2n V G B ⊆ we get: ( ) ( )

The Metrics and Codes in the Additive Channel
Definition 5 (See [6]).The arbitrary subset ≠ .As in the preceding section we define the neighborhood of k -th order of the word n x B ∈ as follows: ( )

NB 1. For the additive channel
Note that the perfect code ( ) V A has maximum cardinality though the convers statement is not always valid.NB 2. Any word from n B belongs to the neighborhood of the first order of only one word of the perfect code ( ) The standard and most used metric in code theory is Hamming's metric [9], i.e. the following function: ( ) It can be taken that this metric is connected with the "natural" basis , , , n E e e e =  in the following way: It is clear that if another basis is chosen, for instance, if , , , n С z z z =  is taken, then another metric will be generated: A more general procedure of metric generation shown above is as follows.For the given subset , , , and the vector n x B ∈ we consider all "expansions" of x into С , i.e. the expression of the following form: and we the following number: x z z ↔ ⊕ = for some i z С ∈ .This relation defines adjacency of vertices and we get a graph, i.e. the set of arcs с Е , which is given by the equality: The distance among the vertices of this graph is given in the standard way: the minimum number of the arcs in the chain connecting these vertices; and the infinity if there is not such a chain. Example.
 is an arbitrary additive channel then the set A generates an MLM norm in n B given by Formula (10).The statement presented below shows that the ability of the code  to correct the errors of the additive channel A can be formulated in terms of the MLM norm generated by the set.
Lemma 4. The code V corrects the errors of the additive channel A iff the following conditions hold: But it follows from (11) that the code V does not correct the errors of the additive channel A .And if ( ) ( ) ( ) The image of any set and the spectra where u , n v B ∈ .For the given MLM metric all the standard definitions of the correcting code theory can be modified replacing Hamming's metric by any basis MLM metric.In particular, the perfect code V with the distance 2 1 + is a partition of the set n B in the union of the spheres of the radius t in the MLM metric.According to (12) the perfect codes in one metric are transformed into perfect codes in another metric.Besides Formulas (12) allow various interpretations of geometrical character.We present two facts which we use further.The preceding statements make possible to build the perfect codes in n В for arbitrary basis metrics if one such code is already built for one basis metric at least.In particular, if

Definition 7 .
(See[8] [9]) The code ( )V A correcting the errors of the additive channel

Lemma 3 .
correspondence to(9).Now choosing the least number of these we define the following norm ([1]; the MLM norm) connected with С : with respect to 9 , , if there are no such representations.The function С is a metric (below, "MLM metric") for any subset n С B ⊆ .In terms of graph theory the described situation is as follows.Let us give the following binary relation on the set of vertices n B : ~i x z norm has the following form: then according to definition: hence, the code V corrects the errors of the additive channel A .Let us discuss an arbitrary basis { }

ρBNB 3 .
and the subset ( ) f A with the basis metric E ρ simultaneous- ly are spheres with the radius t .b) The code n V B ⊆ with the basis metric С ρ and the code ( ) f V with the basis metric E ρ simulta- neously are perfect with the distance 2 1 t + .with different bases can strongly differ the spectrum of distances of the space n If the basis С is chosen as in Example 7 then the preceding statement is equivalent to the following formula:

≠
Consequently, the following variants are possible: adjacency of the vertices are defined as follows.The vertices , have no any simple connection.The situation will be changed to an extent if we consider different MLM norms and introduce limitations on the subject transformations.