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In this work, we study binary linear distinct weight codes (DW-code). We give a complete classification of -DW-codes and enumerate their equivalence classes in terms of the number of solutions of specific Diophantine Equations. We use the Q-extension program to provide examples.

One of the main objective of algebraic coding theory is to classify codes up to equivalence by using a list of invariants. The present work is following this way. We study here a class of linear binary codes whose all codewords have distinct weight and will give a classification theorems. Throughout this work all codes are

linear binary codes. We call an

the Hamming weight

A Hamming isometry of

Two codes

space

a Boolean ring. Furthermore,

a constant-weight code (CW-code) if all nonzero codewords have the same weight. The dual of binary

Hamming codes

weight code (CW-code).

Any permutation of the columns of a k by n binary matrix

Ideally, we would like the rate

rate of a DW-code approch zero very quickly when the code length increase:

in

It is more convenient to use the DW-codes in the construction of other codes by using some technic of construction and not to use it alone.

Definition 1 A linear binary code

the weight mapping:

The simplest example of such codes are the repetition codes. Later we shall give more nontrivial examples. Let

Proposition 2 Let

Proof. Suppose on the contrary that

nation

a contradiction.

Now we give a construction of a

Let

then clearly

generate a

implies that

A generator matrix of

Proposition 3 The

Proof. Since the cardinal of

Let

then

Up an equivalence we have the following result:

Theorem 4 There exists only one distinct weight

Proof. Let

least integer

Multiplying by

consider

basis of

Now we define a linear mapping

is an isometry between

exists a permutation

Example 5

By using the software Q-extension, see [

the unique DW-code

that it is equivallent to the code

rows and then apply the permutation

Theorem 6 Let

have a unique solution which is the k-uplet

Proof.

(1). Assume that

in binary basis, the equality

satisfies the conditions (2).

Conversely, Let

take

condition (2) means that the code of generator matrix

condition (1) implies that there exists an invertible

is the generator matrix of the code

where

uniqueness of development of

Since

Remark 7 Without the conditions (1) and (2), Diophantine equations have

We consider, without loss of generality, that a generator matrix of a DW-code has no zero columns. Indeed, if this is the case, the zero columns are omitted and we consider the obtained DW-code. This assumption is made in the entier paper. We study the automorphism group of DW-codes. We first notice the following:

Proposition 8 Let

Moreover, if

Proof. Clear.

Proposition 9 The automorphism group of any DW-code is nontrivial of even order.

Proof. Let

nonzero. The

We deduce that the dual code

We consider the general case

column, they are the columns fixed by the group

column and then it is clear that

are

Up to equivalence, we can consider that the code $mathcal

Since

The following theorem legitimate the idea of giving a definition to the 3-tuple

We give here the full classification of such a code in several cases.

Theorem 10 If two DW-codes

Proof. Let

We have

automorphism group

So we have

the generator matrix

which is an orbit of the column

and

conclude that the two codes

We have

Theorem 11 If

punctual orbits is equal to the dimension of the DW-code

code of generator matrix

_{1} = d is the minimal distance of

Proof. After a series of permutations and elementary operations on rows of

the first orbit formed only by ones and all other rows are null

generator matrix are zero. Otherwise the first line of another orbit

of permutations and elementary row operations can make null all the other rows of this orbit so

This is a contradiction since two orbits are disjoint. We obtain a generator matrix of an equivalent code denoted

by the same sign

without punctual orbits

It is clear that we have

since

Remark 12 In this case, up to equivallence, each

orthogonal basis:

Example 13 Consider the

Corollary 14 Let two

is equal to their dimension. Then the codes

The converse of Theorem 11 is true under an additional condition.

Theorem 15 Let

then

Proof. Clear.

Corollary 16 The number of equivalence classes of

satisfying the following conditions

Proof. Let the application that maps each equivalence class represented by the matrix

t-tuple

clearly a bijection between the set of equivalence classes and the set of solutions of the Diophantine equation satisfying conditions (1) and (2).

Theorem 17 If

punctual orbits is equal to the dimension of the DW-code

DW-code of generator matrix

Example 18 Consider the

We have

The converse of this theorem is true under an additional condition. Let

columns which are also different from all unitary columns

For each

For all

setting the numbers

Theorem 19 If for all

Let

different way to the choice of

columns we denote by

So we have the following result.

Theorem 20

the number

Example 21 By using the result of the last theorem and the Q-extension software, We show that there exist

Only 4

・ For

So the number of DW-codes with

・ For

So there is no DW-codes with

・ For

So there is no DW-codes avec

・ For

So there is one DW-codes such as

We deduce that there is only four

since

We have necessarily

Theorem 22 If

tual orbits is different from the dimension of the DW-code

with

Example 23 The

of generator matrix

In this case two DW-codes with the same signature are not necessarily equivalent as shown in the following example:

Example 24 Let

We have

We can have two cases

Theorem 25 If

punctual orbits is greater than the dimension of the DW-code

DW-code of generator matrix

Example 26 The

of generator matrix

Theorem 27 If

punctual orbits is lower than the dimension of the DW-code

with

Example 28 The

of generator matrix

Remark 29 Self-orthogonality.

A code which is equivalent to a self-orthogonal code is also self-orthogonal. The property of self- ortho- gonality is then an invariant of the equivalence of codes. We then have the following points:

・ If

・ If

Theorem 30 The automorphism group of a

the group direct product

Proof. Let

For

Clearly the

Now we are going to show that

If

Let

Now let

defined by

Example 31

・ Consider the

of generator matrix

・ Consider the

of generator matrix

The authors would like to thank the refrees for their helpful suggestions and remarks.