Classification of the Subsets nB , and the Additive Channels

The problem of classification of the subset of the vertices of the n-dimensional unit cube in respect to all “shifts” by a vector from n B is studied. Some applications for the investigation of the additive channels of communication are represented.


{ }
0,1 F = be a two element Galois field and 2 n F be an n-dimensional space on that field.In other words, 2 n F is the set of vertices of the n-dimensional unit cube, { } 0,1 .
n n B = The subsets n B have many different interpretations in the terms of Boolean function theory, or of correcting code theory, or of partially ordered set theory, or that of additive channels etc.And each of these theories is connected with a certain class of restrictions imposed on the properties of the subsets, n B We consider the "shift" of the subsets n B , and we define equvalence as equality that is accurate within the shift.To define the subsets stabilizers and the transitive subfamilies we use the classic ways connected with Burnside's Lemma.

A y y x x A + = + ∈
Thus A y + is the shift of the set A on the vector y .The transitive set generated by A has the standard form: ( ) { } . .
It is well known [1] [2] that A G is a subsets in n B and the cardinality of the transitive set ( ) where ( ) is the index of the group n B in regard to the subsets A G .Example.

( )
G A it is sufficient to know the cardinality of the stabilizer A G .Let us note that the group A G acts on the given set A , that is, A G is a stabilizer and we can use the same way of argumentation as we did above.
If x A ∈ , then the transitive set ∈ is defined in the standard way and: where x G is the stabilizer of the element, x A ∈ .Taking into account that: , for all x .Then we have from (2): is the power index of the prime number p , which is included in the canonic presentation n , then the following statements hold true: Corollary 1.The following inequality holds true: Lemma 2. The stabilizer Proof.We assume, 0, .
, , , m A x x x =  satisfy the following system: Adding up all the equations of the system, we get the following equality: .
which is a contradiction and it proves the Lemma.
In the general case, if the element y belongs to the stabilizer A G of the subsets , , , m A x x x =  , the following holds true (according to the definition): Let m S be a symmetrical group of the degree m .We denote the elements of the group m S corresponding to transformation (3) by y g .Consequently, the element y g should by written as follows: Proof.If ( ) , we have from ( 4): It follows from (5) that: ( ) To calculate the stabilizer one has to consider the multiset:

{ }
; , , where , which has a key role for the further considerations.Let , , , m A x x x =  , and: where ij α is the multiplicity of the inclusion of the element, ( ) Lemma 4. The stabilizer A G , of the set A , is the sets of elements ( ) , where each occurs m times plus the zero element.
p q pairs: ( ) x x and ( ) x x , have no common elements; otherwise they coinside.
Thus, the set of pairs ( ) form the partition A , and the point y belongs to A G , according to Lemma 3. From Lemma 4 a simple algorithm for building the stabilizer A G follows and, as a matter of fact, it is re- duced to building of the multiset, A A + .Complexity of such an algorithm is ( ) The volume of the input information is the length of the recording of the set A , that is, ( ) O mn .

Lemma 5. If the cardinality of the subsets A and that of the stabilizer A
G satisfy the following conditions: For any 1 x A ∈ we build the set { } x A A ∈ ∩ .Then the vector x can be represented in two ways, namely: x x y y A = + + ∈ , which contradicts the choice of the element 2 x .Hence, the following holds true: we have: We denote x A ∉ , we have: A z G ∉ .It follows from ( 6) and (7) that x A ∀ ∈ is represented either in the form: x z x y z x y A + = + + = + ∈ .It can be proved in the same way that x z A + ∈ , for the case, 2 2 x x y = + , Consequently, A z G ∈ We got a contradiction and it concludes the proof of Lemma 5 if we take into account Lemma 1.

Lemma 5 is a useful tool for calculation of the stabilizer
. Its content can be interpreted as follows.If it is possible to define 2 2 1 k − + elements belonging to A G , then, taking into account that the cardinality of a stabilizer is an exponent with the base 2, we directly get: Taking Lemma 4 into account, we get: , , , A x x x x = Consequently: { }  x x x x + = + then the following equalities hold true:

. x x x x + = +
Consequently, the following statement holds true:  N y is the set of the (stationary) points y of the transformation, that is: of the following equation: Proof.According to Lemma 3, Equation ( 8) is equivalent to the system of the following equations: where the partition is chosen for the sake of certainty.Let us note that the following equation: and it does not depend on y if 0 y ≠ .Indeed, choosing an x we get: z x y = + .Further, if ( ) , x z and ( ) , u v are two solutions of Equation (10), then either these solutions do not overlap, or they coinside.Indeed, we get 0 x z u v + + + =, from x z y + = and u v y + =; consequently, it follows from x u = that z v = In the same way, if x v = , then z u = Thus, all the solutions of system (9) can be obtained by choosing m pairs from 1 2 n− pairs, which are solutions of (10).Theorem 1.The following equalities are valid: Proof.We get from Burnside's Lemma: , taking into account Lemma 7, we get: ( )  For 0 y ≠ .This directly proves Formula (11).For the case 2 1 k m = + , taking into account Corollary 2, we get: ( ) 0 N y = for all 0 y ≠ , which proves formula (12).Thus, the above statements make, more or less, possible to know the structure of the stabilizer A G of the set Let us also note that, according to Corollary 1, . On the other hand, as Example 1 shows, for any subgroup 2 and for any collection of contiguous classes 1 , , For an odd ( ) t q + , and its stabilizer M G has 2 t elements.This shows that it is possible to draw the above mentioned boundary for the stabilizers of the considered sets.The following example of a contiguous class is not so bad evaluation for the cardinality of the stabilizer of the set A .The "average" value of this boundary in the whole interval of the cardinalities 1, 2 n     , is 2 n and this can serve as a "realistic" boundary for the cardinality of the stabilizer for a uniform distribution on the family of the sets ( )

∑
Shifts and Additive Channels.One of the applications of the above considerations are the so called additive channels.
We call any subsets Thus, any word x , if transmitted through the additive channel A , is transformed into one of the words x′ of ( 14), in the result of the shift by the vector i y .
Definition 1 [5].We define the k th order neighbourhood of the vector, , corrects the errors of the additive channel where , 0, The equivalent definition has the following form: The code , where , 0, , , 0, , .
As the k order cardinality does not depend on the vector v we denote: ( ) Let us note that for the cardinality of the code V correcting the errors of the additive channel Actually, condition (15) makes possible to decode the initial message at the channel output through a standard "decoding table" of any word.
If one takes the sphere of radius t with the centre at zero as A , then he gets the classic channel through which there take place no more than t distortions of the form: 0 1,1 0 → → .The main problem when investigating a given additive channel A is the building the code V of the maximum cardinality, correcting the errors of the channel A .Consequently, each additive channel generates its own coding theory, and the possibilities of examining and sorting out all these communication tools are rather limited.At the same time, some most simple considerations show that many of these additive channels are equivalent (identical) in the sense of their content.Indeed, the channels, A and A y + , are equivalent for any n y B ∈ , in the sense that any code V ,correcting the errors of the additive channel A corrects the errors of the additive channel A y + as well, and vice versa.The above classification of the additive channels is based on these considerations.In particular, one can always consider that ( ) 0 0  belongs to the channel A otherwise one could pass to the equivalent channel including the zero vector, without any loss of generality.
Another definition of equivalence of additive channals is directly connected with the error correcting code.Let

( )
, Х A V be a predicate given on the Cartesian product 2 2 n n B B × or:

B
of all m-element subsets of the cube .nThe transformation group n B operates on this set as follows.For any a transitive set is found in terms of the stabilizer A G of the set A :{ }; be an arbitrary cosets to the subgroup A ; then .element of the group, A , belongs to the stabilizer of the set M , and thus: This example will be used in the sequel.As (1) shows, to define the cardinality of the transitive set

Lemma 1 .
equal to the index of the unit subgroup E, or: The following comparison holds:This immediately follows from the formula of the partition A :

(
Now let us calculate the number of the sets that are transitive in regard to the group n B The tool for such calculation is Burnside's Lemma:[1] [2].Lemma (Burnside's) 6.The number n m L of the equivalence classes or transitive sets is as follows:

,
the number of the transitive sets n m L which are generated by the action of the group


, then, according to Theorem 1, we have for the numbers n L of the transitive sets the fo- llowing equality:Corollary 4.
All the partitions into pairs of the set A are generated by one of them, for instance: