**Open Journal of Discrete Mathematics** Vol.3 No.3(2013), Article ID:34511,6 pages DOI:10.4236/ojdm.2013.33025

Dirichlet Regions and Perfect Codes in Additive Channel

BIT Group, Moscow, Russia

Email: garib@firmbit.ru

Copyright © 2013 Garib Movsisyan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received March 13, 2013; revised April 15, 2013; accepted May 17, 2013

**Keywords:** Dirichlet Regions; Perfect Codes; Additive Channel

ABSTRACT

In the present work, the class of metrics connected with subsets of the linear space on the field, GF(2), is considered and a number of facts are established, which allow us to express the correcting capacity of codes for the additive channel in terms of this metrics. It is also considered a partition of the metric space, B^{n}, by means of D-representable codes. The equivalence of D-representable and the perfect codes in the additive channel is proved.

1. Introduction

We consider the additive channel of communication [1-4] as a transformer of information, which is a generalization of the classical binary channel with limited number of distortions, Many notions and facts in the present work take their origins in the classic coding theory and are the direct analogues of the well-known results [1-5].

The “noise” generated by the additive channel leads to the fact that there appears a word at the outlet of the channel which is different from that at its inlet. In connection with this there rises a necessity of transforming (coding) information for conducting it through the given channel, as well as a necessity of retransforming (decoding) it at the channel outlet. This circumstance makes one introduce such standard notions in the coding theory as: error correcting code; transfer/decoding speed, etc.

On the other hand, as there are many additive channels, the problem of ordering and classification of such channels rises, taking into account the main difficulty, namely, the possibility of correcting the generated errors.

We consider the class of metrics connected with the subsets of the space on the field, GF(2), and establish a number of facts which allow us to express the correcting capacity of codes for the additive channel in terms of this metrics. Also, we consider the partition of the metric space, , through -representable codes. The equivalence of -representable and perfect codes in the additive channel is proved.

2. Codes in the Additive Channel

Let be a Galua field of two elements and be an n-dimensional vector space on that field.

If is a subset of, then the notion of the additive channel is connected with A, as follows.

Any of the vectors, , in the channel, , is transformed into one of the following vectors:

where is the operation of addition (with respect to mod 2) in the space,.

Definition [1]. For any vector, , we call the neighbourhood of order t with respect to C, the following set:

For convenience we take,.

Examples:

1) Let. Then, where is the logic negation of.

2) If is the parity counter, then:

where is complement of the set, , in.

3) If, then in the additive channel, , can occur no more than t ‘errors’ of the form, Consequently, is a sphere of the radius, t, having its centre at the point, x.

Thus, we get the classical case of the binary channel with limited number of errors.

Definition [2]. The code, , corrects the errors of the additive channel, , if:

An equivalent form of this condition is:

or the one symmetrical to it:

It is obvious that the preceding definitions are symmetric with respect to the pair, , and therefore, both “generation” and “correction” of errors have the same nature.

Statement 1 [3]. If the code, , corrects the errors of the additive channel, , then the code, , corrects the errors of the additive channel,.

To describe the “relations” of the additive channel, , and the code, , correcting the errors of that channel it is convenient to introduce the following double-case predicate,:

Definition. We call any pair, , additive if it is a solution of the following equation:

Taking this definition into account, the property of perfectness of codes can be written as follows:

Note that there are as many additive channels, as there are Boolean functions, and a few of them do not essentially differ from each other. It is not clear how to classify such channels yet, but the following statements correspond the commonly accepted viewpoint.

Definition [3]. The channels, A and C, are called equivalent if any code correcting the errors of the additive channel, A, corrects the errors of the channel, C, and vice versa.

Introducing the following relation of partial order, one can formally write:

If, then, which is natural.

This property makes possible to look additive channels with the ‘best’ and ‘worst’ correcting capacities for each

Statement 2 [3]. The additive channels, and, are equivalent for any.

Statement 3 [3]. If, then.

It follows from the preceding statements that one can consider—without loss of generality:

a) If is a class of additive channels equivalent to, then it is sufficient to solve the coding problem for any representative of that class.

b) The additive channel, , includes the null vector, which can be interpreted as the possibility of errorless transfer of the signal through that channel.

As it follows from that, then an analogical statement is correct for the vector, , too, i.e. it is sufficient to discuss the codes including the null vector.

Thus, it follows from that the sets, and, can overlap only at zero, and the search of the code, , is to be organized in the set, Below:

As the power of the neighbourhood of order t does not depend on the vector, , we make the following denotation,

Note that for the additive pair, , the following limits take place [2]:

It is clear that the upper limit is reached for the perfect pair,

3. Metrics and Codes

The standard and most used metrics in coding theory is Hamming’s metrics, i.e. the following function:

One can consider that this metrics is connected with the “natural” basis, , in the following way:

It is obvious that if another basis, , is chosen then another metrics is generated:

The more general procedure of metrics generation in the above-mentioned way is as follows. For the given subset, , and the given vector, , we consider all expansions of x with respect to, i.e. the expansions having the following form:

(1)

and then we put the following number into correspondence with the above expansion:

Then, choosing the least of these numbers, we define the following norm (the MLM norm) connected with:

Lemma 1 [4]. The function, , is a metrics (below, MLM metrics) for any subset,.

Example:

4) If, the MLM norm has the following form:

In particular, for, the MLM norms of vectors in take the following values:

In terms of graph theory, the above situation is as follows. We give the following binary relation on the set of vortices,:

This relation defines adjacency and, geometrically, the set of vortices of the N-dimensional unit cube corresponds to the metrical space, and the distance between two points in is equal to the minimum number of sides in the circuit connecting the corresponding vortices, or it is infinity, if there is no such circuit.

Let be a basis for the additive channel, A. We consider any basis, , in the space, , where, and f is a linear reversible transformation, , defined in the following way:

We denote the image of the set, , by:

It is obvious that if, then for all.

We denote the linear shell of the set, , by, and we denote by the subset in satisfying the following condition:

where + means the direct sum the subspaces. It is obvious that, if.

The following holds true.

Lemma 2 [4]. For any vectors, , the following relations hold true:

Lemma 3 [6]. The code, , corrects the errors of the additive channel, А, iff the following holds true:

Example:

5) We consider the channel,

, as an illustration. “Physically”, the channel A means that the “errors” of the form, , which take place either in the 1^{st} place, or in the 1^{st} and 2^{nd} places simultaneously, and so on. Thus, and to build a maximum volume code correcting the errors of the given channel we use Lemma 3. It is sufficient to consider all the subsets, , for which:

Let

Lemma 4. For any holds true the equation:

where.

Proof. It follows from the conditions of lemma that the vectors, , are one to one represented in the form [7]:

where.

Consequently:

(2)

We assume—without loss of generality—that:

Then:

where:

Consequently, taking (2) into account, we get:

Q.E.D.

Lemma 5. For any additive pairs, and

, where the pair,

, is additive, too.

Proof. Let for hold the following equation:

(3)

From the definition of the direct sum of sets it follows that:

where

Consequently, taking (3) into account, we get:

That is, the pairs, and are not additive.

Q.E.D.

4. Partition of the Metric Space into Dirichlet’s Regions

Let be the metric space with an MLM norm, , where is a subset in.

We define for any the Dirichlet region, , of the point, , in the following way:

It is obvious that:

(4)

In fact, Dirichlet’s region of the point, , includes all points of the metric space, which are not farther from than from the other points in.

It is easy to notice that it is sufficient given the coincidence of the sets, and, for (4). Nevertheless, this condition is not always necessary, which can be seen from the following example.

Examples：

6) Let

Then we have:

that is,

7) Let

then:

It is obvious that: and do not overlap in pairs. It follows from this example that the condition, , or is not necessary for the equality (4).

The following theorem ‘connects’ and, giving the answer to the question: which are the conditions providing Equation (4).

Theorem 2. The equation:

holds true for all iff:

Proof. Taking (4) into account, it is sufficient to prove that, i.e. for every vector,

there is such that. As for any vectors,

holds the following:

then iff

Q.E.D.

Definition. The code, , is called -representable in the metrical space, , if all Dirichlet regions of the points in do not overlap in pairs.

We note that D-representability of a code is connected with a certain metrics and, in general, this property does not preserve if the metrics is changed.

Example:

8) Let:

We have in the metric space, that

. But in the metric space,

Theorem 3. The code, , is D-representable in the metrical space, , iff the code, is D-representable in the metrical space, , for all.

Proof. For we have:

then it is logical to consider the case,

Necessity. Let the code, , be D-representable in Then the following holds true: for every vector, , the vector, belongs only to one Dirichlet region, where. Then, according to Theorem 2, we have that

Consequently, the code,

, is D-representable in.

Sufficiency. Let be not D-representable in

. Then there is such vector, , that

, where

, and

. The following cases are possible:

a)

b)

c)

d)

It is not difficult to prove with these that in the space,

, including the vector, , the code,

, is not D-representable, which is a contradiction. Q.E.D.

This theorem can be formulated in another way.

Corollary. The metric space, is partitioned into Dirichlet’s regions of the points of the code, , iff the metric space, , is partitioned into Dirichlet’s regions of the points,

, for all.

We describe the algorithm of building the code, , partitioning the metric space, into Dirichlet’s regions.

Step 1. We choose an arbitrary code, , from the space, , partitioning the metric space,

, into Dirichlet’s regions for which

.

Step 2. The code obtained through the formula:

partitions the metric space, into Dirichlet’s regions.

Let any basis of the channel,.

Theorem 4. If the code, , is -representable in the metric space, then the additive pair, is perfect for.

Proof. It follows from Theorem 3 that the code,

, is -representable in

. As M is a basis for, then it follows from Theorem 4 [8] (taking Lemma 2 and Lemma 3 into account) that the code, , is perfect in. Applying Lemma 3 and Lemma 4, we get that the additive pair,

, is perfect for every vector,

.

Q.E.D.

Nevertheless, we note that the perfectness of the additive pairs:

is not a sufficient condition for -representability of the code, , in, though is a basis for all.

Example:

9)

The code, , is -representable in, but it is not -representable in, though the additive pair, , is perfect in.

Theorem 5. If the additive pair, is perfect, then V is -representable in the metric space,

Proof. As for every vector, and

, then. On the other handfrom perfectness of the code, V, it follows that the set, , for all disjoints. That is, V is D-representable.

Corollary. The additive pair, (is perfect if the metric space, , is partitioned by the Dirichlet’s regions, , of the points, , from.

Thus, we reduced the problem of building the perfect Pair, to the problem of -representability of the code, , in the metric space,

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