^{1}

^{2}

^{3}

In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space
*B*
^{n}
are presented. Also, sums and multisums of various “classical figures” as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation
X
+
Y
=
A
, where
, are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.

If

where

This addition operation for members of

In other words, if

Thus, the sum of subsets

Examples.

1. if

2. if X is a subset in

3.

Also,

The family

The following inequality is valid:

Both limits are achievable here. The following statements describe the sets in which these limits are achieved.

Definition [

Statement 1. The upper limit is achieved if

Corollary. If

We consider an arbitrary subgroup

where

Definition [

Statement 2. The lower limit is achieved if there exists

Corollary. If

Now let

Example.

1. If

In this case X has a non-obvious stabilizer if all constituents of X can be partitioned into the pairs

In the general form the stabilizer

Statement 3. The constituent

This statement can be obtained by analogical consideration for

From the above statement one can construct the following algorithm for building the stabilizer

1. First we build the multiset

2. Then we choose all the pairs in C having the multiplicity m.

3. Then we build all partitions in A out of these pairs.

4. If

Example.

1. Let

Then:

This means that all pairs

The sum of the pairs in each of the solutions is the same. Hence, the following set:

is a stabilizer for X.

Below we present the simple properties of the operation “+”―it is addition in the sense of Minkowski, as was mentioned above―which can be taken as properties of an algebraic system with basic set

1. Assosiativity:

2. Commutativity:

3. Distributivity with respect to union:

4.

There are finitely many other relations connecting constituents of the algebraic system described above.

Let

Statement 4 [

Statement 5.

Here

Proof. Let us note that if

then for

or:

and if

Example.

1. We consider the sphere

Formula (2) in the preceding statement allows the following generalization connected with addition.

Let

Statement 6 [

Corollary. For

1.

2.

Statement7 [

for

and the next one is valid:

for

Corollary. For

A facet, or sub-cube, or interval in

where (≤) is a coordinate-wise partial order relation in

In other words, an interval can be given by a word of the length n in the alphabet

Indeed, if:

then the code

Let

Examples.

1. If

2. If

If

Let

If the operation “⋆” is introduced in the alphabet A by the following Caley table [

then the sum of the intervals J of the system defined above as a sum of subsets is the interval the code of which is calculated by the codes of items (addends) using the above Caley table.

Statement 7 [

Examples.

1. If

On the other hand, we get by definition:

i.e.

2. If

Statement 8 [

Thus, the distance between the intervals

Let

By definition

if

Hence, g permutes the coordinates of the point , leaving its Hamming weight unchanged.

At the same time the relation

Thus, each layer

Let

Statement 9 [

For not large values of the layer the following table of addition is valid:

Note that Formula (3) can be rewritten for any number of terms, using the above-mentioned property of distributivity.

Indeed, using (3), we get:

which makes possible to use (3) again.

Example.

1. Let us find the sum

NB. As each layer

If we take subspaces in

in terms of cardinality:

Thus, “theory of addition of subspaces” being a well-developed part of linear algebra, makes possible to answer many questions concerning the subject problem.

The k-dimensional interval we denote by

According to statement 6, we have:

i.e.

Let

Statement 10.

For the cardinality of the set

Corollary.

Proof. If

if

Therefore,

Analogous to the preceding statement and corollary we get the sum of the sets

Statement 11. The following relation is valid:

Corollary. The cardinality of the set

Statement 12. The following is valid:

where

Proof. We have from statements 9 and 6:

Q.E.D.

Let

The ‘simplest’ equation in sets is as follows:

where

It is clear that Equation (4) always has the trivial solution

Examples.

1. If

2. If A is a subspace of

3.

Now, let:

Then

is expressed by the norm of the sum of these solutions.

On the other hand, if:

then

that is,

Thus, the set

In an additive channel of communication [

Let

We introduce the following parameters:

Introduction of such definitions as

Then, for the minimal and maximal cardinality set

It is not hard to prove that:

As every solution

Statement 13. If

Statement 14. For the subspace

(b)

Proof. It follows from (5) that it is sufficient to prove for (a) that:

Let

On the other hand, it follows from Statement 13 that

The proof for the case (b) is analogical.

Statement 15. The following estimations are valid:

1.

2.

3. If

Proof. Items 1 and 2 of this statement were proved in [

Necessity. We assume that:

and that the pair

It follows from (7) that:

According to the statement, we have that the set

For

Sufficiency. Let

As

The statement is proved.

Examples.

1. The pair

2. The pair

If we keep to these examples, then we can assume that there exists some monotonous dependence of the function

Corollary. For the halfspace

The “seemingly obvious” hypothesis that the upper limit of

Examples.

1. Let

2. Let

We have:

Statement 16 [

Consequently, the problem of constructing of an equidistant code with the distance k having the minimal cardinality can be formulated in terms of solvability of the equation

Definition. The set

is solvable.

It is clear that a quadrate always contains the zero point.

Example.

1. If A is a halfspace in

The notion of ‘quadrate’ is connected with problems of equivalence of additive channels [

Let:

We denote by

Statement 17.

From this and taking into account the known estimations

Statement18.The following inequality is valid:

At the same time equality takes place if there exists a perfect code in

Consequently, the problem of constructing the code of maximal cardinality ? in particular, a perfect code ? is reduced to finding the solution of maximal cardinality for Equation (10) among all quadrates of the union of layers

Statement 19 [

Corollary. The preceding statement is valid for any number of summands.

Now let

Definition. The set of matrices

At the same time, if

Statement 20. Let

The second definition of addition of sets from

Definition. A multisum of two sets

in which each member

Examples.

1. If

2. If

It is clear that by definition

In particular, the following expression is valid:

where C is an arbitrary subset in

It follows from this that:

Let

Statement 21. For the multiset

where

Proof. Let z be any member of the multiset

We assume (without violating generality) that:

Hence, we have:

that is:

From this, taking into account Statement 9, we get Formula (12).

Corollary. For

а)

b)

if

c)

if

Statement 22. For the multiset

where

is the multiplicity of the members of

Corollary. For

a)

b)

Statement 23. For the multiset

where

is the multiplicity of the members

Corollary. For

Statement 24. For the multiset

where

Corollary. For

Statement 25. For the multiset

where

is the multiplicity of

Corollary. For

Statement 26. For the multiset

where

Finally, we define the operation “/”, that is, subtraction for multisets.

Let

Definition.

Example. We consider the multisets:

From Statements 22 and 12 we get:

Vladimir Leontiev,Garib Movsisyan,Zhirayr Margaryan, (2016) On Addition of Sets in Boolean Space. Journal of Information Security,07,232-244. doi: 10.4236/jis.2016.74019