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In the present work, a construction making possible creation of an additive channel
of cardinality s and rank r for arbitrary integers s, r, n (r≤min (*n*,*s*-1)), as well as creation of a code correcting errors of the channel A is presented.

We consider an additive communication channel introduced in [

The “noise” generated by an additive channel leads to a word at the exit of the channel which differs from the transmitted one. This circumstance makes one to find the leads to creation of necessary initial prerequisites for introducing standard notions of an error correcting code in the coding theory, as well as the notions of the speed of communication, decoding etc.

Thus, the problem of constructing new codes from known ones has certain interest for coding theory. In this work, using certain combinatory constructions, some new codes for additive communication channels are constructed (also see [

Let

If

Any vector

here

Definition 1. The following set:

is called the t-order neighborhood with respect to A of any vector

As the cardinality of the t-order neighborhood does not depend on the vector x, we use the denotation:

Definition 2. The code

An equivalent writing of this condition has the following form:

or here is another one which is symmetrical to the preceding one:

Below, without loss of generality, we take:

Let us note that for the cardinality of the code V correcting the errors of the additive channel

Besides, the code V for which the upper limit is reached is called the perfect code correcting the errors of the additive channel A.

To describe ‘interrelations’ of the additive channel A and the code V correcting the errors of this channel, the following convenient two-place predicate X(A, V) is introduced:

If the cardinality of the channel A is fixed, then there exist

here

The Hamming metric is a standard and mostly used metric in theory of coding, defined by the following function:

One can accept that this metric is connected with the ‘natural’ basis

It is clear that choosing another basis

A more general procedure of metric generation is as follows. For a given subset

And for each such representation, we juxtapose the following number:

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

The function

Let

We denote the image of the set

It is clear that if

According to [

Lemma 1. The image

Theorem 1. For all

Corollary 1. If

This corollary can be paraphrased as follows.

If

Anyhow, one cannot assert that the condition

Example 1. For instance, if

For Hamming metric the proof of this fact can be found in [

Theorem 2. The non-trivial perfect codes, correcting the errors of the additive channel

(a)

If

Lemma 2. The code V corrects the errors of the additive channel А, if the following conditions hold true:

Now we consider the set

Theorem 3. The code

Proof. Taking into account Lemma 2, it is sufficient to prove that the following inequality holds true:

Let us consider two cases:

1)

Then it is not difficult to prove that

where

As

2)

Then

The theorem is proved.

Without any loss of generality we can take

Applying Theorem 3 sequentially to the pair

where the vector

The code

From this we have:

Now, the proof of the following theorem is not difficult.

Theorem 4. If

Corollary 2. If

In other words the communication speed for the pair

Corollary 3. For any integers,

Proof. To prove this statement it is sufficient to apply Theorem 3 to the pair

It follows from these that

Q.E.D.

Example 2. Let us consider the additive channel

It follows from Lemma 1 and Theorem 2 that the code

Applying the above-described method (Theorem 3), we get the channel:

And the code

where

The following holds true for the constructed pair:

It follows from here and Theorem 1 that the constructed code

Let us consider the partitioning of the set

It follows from the preceding theorem that for arbitrary

Example 3.

Let us again come back to the definition of the perfect code. The standard definition of the perfect code means that it is a set correcting the errors of an additive channel in the MLM metric in which the upper limit of the cardinality of the code is reached. Such a definition provides fixation of the code cardinality, leaving wide room only for maneuvering for its geometrical form. But the definition of the perfect code correcting the errors of the t-order neighborhood (for Hamming metric, correcting the t-multiple errors) means partitioning of the space

It is obvious that there is a “geometrical sense” in the second definition, which is strictly definite, stating the t-order neighborhood (that is, the multiplicity t of an error for Hamming metric). The parameter t defines the neighborhood uniquely (a sphere of the radius t) and, consequently, the cardinality of the neighborhood as well,

which equals

Taking these considerations into account, one can conclude that these two notions do not always coincide. To demonstrate this fact, let us discuss the following example.

Example 4. A perfect code in the ‘geometrical sense’ does not exist for

Consequently,

where

It is clear that the channel

Garib Movsisyan, (2016) Construction of New Codes from Given Ones in an Additive Channel. Journal of Information Security,07,165-171. doi: 10.4236/jis.2016.73012