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