Partition and the Perfect Codes in the Additive Channel

Many problems of discrete optimization are connected with partition of the n-dimensional space into certain subsets, and the requirements needed for these subsets can be geometrical—for instance, their sphericity—or they can be connected with certain metrics—for instance, the requirement that subsets are Dirichlet’s regions with Hamming’s metrics [1]. Often partitions into some subsets are considered, on which a functional is optimized [2]. In the present work, the partitions of the n-dimensional space into subsets with “zero” limitation are considered. Such partitions allow us to construct the set of the group codes, V, and the set of the channels, A, between the arbitrary elements, V and A, having correcting relation between them. Descriptions of some classes of both perfect and imperfect codes in the additive channel are presented, too. A way of constructing of group codes correcting the errors in the additive channels is presented, and this method is a further generalization of Hamming’s method of code construction.


Introduction
Let   0,1 В  be a Galua field of two elements and be a linear vector space on that field.We consider the family of the subsets, , satisfying ; ; (summation is with respect to ). 0 The first three properties are usual for partition of the subset, , and the last "non-zero" one reflects the specificity of the further usage for constructing of the correcting codes.The case, , is particularly important, because it leads to constructing of the perfect codes.Below, the term, partition of the set, , , , , is used in the sense of (1), i.e. it is the partition into "zero" subsets.
The problems of existence, constructing and partition of for the given 0 1 , , , k s s  s and п have not only combinatorial-set interest, but also that in connection with correcting code construction.It is worthy to note that in correcting code theory the decoding regions form partitions of the space, , if decoding region pairs do not overlap each other.Consequently, some code classes-particularly, the perfect codes in the additive channel-make it possible to construct the partitions, 1 0 0 1,3,3,3,3,3 , , , , , is the partition of the space, , if: 5 1 1 0 0 0 0 1 1 0 1 0 0 , 0 0 0 1 , 1 0 0 0 0 0 1 0 С  is the partition of the space, , if:

n B
From the partition, , of the set, , one can obtain the partition, , , , k C C  We present (without proof) the following lemma that describes some trivial properties of the partition, .
, , For every the following takes place: a) ; s partitions of the subset, . where: , in the form of the matrix: For every pair, .We define the sets, , in the following way: a) For every C , contains only one element of any line and any column of matrix (3), and it satisfies the following con-dition: no pair of all the sets, p ij C , is overlapped and every one of these sets has the power, s; that is: Let us consider the set: , 0, 0, .
we have: (summation is with respect to ), mod 2 and as: , , , , We consider where , and is defined in the following way:


It is easy to prove that   s is an integer, and , according to the assumption, there exists the following partition of the space, : Now we are going to describe the construction of the group code set algorithm and that of the channel sets, using the partition of the set from the ND space into "zero" subsets.It is proved that any code of the constructed set corrects all errors of every additive channel in the set of the respective channels.
An additive channel is given by the set of vectors of errors, , , , , where is the initial vector, , and is the addition operation with respect to [3].
The neighbourhood of the order of t of the vector, , with respect to is defined in the following form [4]: The code, V, corrects the errors of the additive channel, where , , Classical boundaries of Hamming and Varshamov-Gilbert for the power of the code, V, correcting the errors of the additive channel, A, have the following form [5]: The main task for the given channel, A , is the con- struction of the maximum volume code correcting the errors of the channel, A .
n The code V is called perfect for the additive ng condition is satisfied: The code, , is called quasi ch , the code, V , is perfect for the channel, A C  .code, , fo In o r words, the quasi-perfect V the r the channel, A , satisfies the conditions: 1) , , , ; We denote by   , , n l A the group code, from , of th n B e order, l , correcting all the errors of the additive channel, A .
We def th ine e product of the Boolean matrix, , or the dimension, т п  , and the vect or, , in the follo way: wing is called checking for the code,   , , n l A , if for all vectors and only for them the following equality takes place: code T 0, Нx  where all operations are carried out with respect to mod 2 V, correcting the errors of the additiv ( [6]).
To build the code, e channel we use the following construct connected with the partitions presented above.First we build the additive channel, then the group code correcting the errors of that channel.
We consider the ma be the negative integers and there be ng form: , where , \ , , , .
Here is the unit matrix of the order, E i s , and E is , where the logic negation of E .We build the channel, , where , and is from all Example 4. We build a channel for the case: x given in the fo an matri llowing way: Using the definitions of the numbers, , and the ve i n ctors, i x , we obtain: of such channels has the following power ,where , otherwise It is obvious that and the above described channel, , other , depends on the choice of the vector, , from , and the checking ma , , , defines the code, one to one; consequently, the set of 0 1 , , , , , , , 0,1, , , has the power: , and the additive channel, We prove that the group co aving de, , h H as its checking matrix corrects all errors nnel, of the cha To prove th th is it is enough to show at for any А  , x y  takes place: . It is easy to show that: Hence, taking into account that , coincide with those of matrix (5), where the block, , is located, we obtain: x y , are the lines of ma x (5).Then we obtain from (7): Hence, taking (9) into account, we obtain tha exist such vectors,  t there We obtain, ma 1: is a column of matrix ).Then we have from (7): ence, taking (9) into account, we obtain that there exist the vectors, 1 , Again, applying Lemma 1, we get that for any vectors, ,if 1, 1.
. As a result, we have that every co , corrects the errors of any channel, , satisfies the condition (4), that is, it is perfect.In result, we get the following statement., , , , , , , , corrects the errors of any and it is perfect., then the above described method of codes is the Hamming me nel , n C C  building of group thod of group codes correcting the errors of the chan-,   1 0 ,0 10 ; 1,2, , 1 in th ho e above described algorithm of const of channels aking into account the following conditi ructing the set , t on: \ is an odd number .
We build the set of channels, in the foll ing way; any channel, being of all lines of the Boolean ma-tri ng way: , otherw It is obvious that the above described procedure of constructing uniquely defines the set,   .
C e orollary 3. Th perfect code,

, if of the space, В
A is the zero chann Example 5 el. .We consider the partition,

C
where: Consequently, the corresponding perfect code: , 000000 , 1001 , 010101 , 110011 corrects the errors of the zero channel, A , of the fol- lowing form: . As In result, we get the code, for   are zero sets and they partition, partition the n get the followi the above partition i , Corollaries 1 a sequence of the par rfect codes, as We have from , ,  ,  C C CC we ca ng perfect code and the titions of the space and, the well.
Which is the checking matrix of the perfect code,

 
15,11, А , where the channel, , has the form: Example 7. We use the ion, partit C as in the preceding exam- ple and we build the 1 2 3 , , Then we build the matrix,   which is the checking matrix of the code, where A is one of the channels in the set,   0,3 A or .F instance: В , and for some integer, 3, , , , , , , ,  the group code with the checking matrix, Example 8. We consider the partition, Then we build the matrix, Which is the checking matrix of the code,   14,10, А , , . s s i k   win Now we are going to consider the case, The interest in this case is due to the follo stances.According to Theorem 1, existence of a partition in depends only on the parameters, n and s, and this si lifies the algorithm of both code and communication described in §2.Besides, in the case,  , nnels is sim-classification of building both codes and cha plified as well.
Theorem 3. If a divisor of and The proof is similar to the one for Theorem 2. In the following two examples, we build two different channels and the codes corresponding to both, using the pa 2 1 ,0 Example 9. Using the partition, 1 1 0 0 0 0 1 0 0 1 0 0 , 0 0 0 1 ,

Addenda Z
is the regular nullmatrix.
2) If 2 m  , then there doesn't exist a regular nullmatrix.
3) If 3 m  , then the following matrices are regular null-matrices: 2 where 2 С is any regular null-matrix of where is an where is an arbitrary null-matrix in We introduce in m F  partial order, requiring:

Examples. 6)
We describe all extreme matrices in .n we can take one of them away, and the lines in t obtained matrix will again be different, and this m hat .umns emen h of the columns of the matrix, , has even number of units..This follows the fact that the matrix is a null-matrix.The significance of the intr d definitions and the above results is that they make possible to obtain any matrix just adding some null-matrix to .This is due to the fact that taking away column any matrix, ilding deadl ven matrix, C [7].


; any vector, , at the exit of such a channel has the form: and the column numbers of the sub-matrix, T ,11, А , where the channel, , С , of the space, B 4 , we build from of the present paper we consider the group perfect codes built through the partition,

2 С y regular null-matrix n 4 F
m , there is neither regular, nor unit col in the m rix, C .ontyev, G. L. Movsisyan and J. G. Margaryan, of N-Dimensional Space on GF(2) into Diri- , taking into account that the necessary condition of their existence is the evenness of the number, s , if 0 s The following construct of the direct product allows building new partitions out of the given ones: is odd.