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.
Distinct (Constante) Weigth Code Automorphism Group Extension Theorem of MacWilliams Diophantine Equations1. Preliminaries
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 -binary code every dimensional subspace of. Recall also that
the Hamming weight of vector is defined to be the number of nonzero components of. The mi- nimum of weights where is the minimal distance of the code.
A Hamming isometry of is a linear application such that, for every
. It is well known that in binary case, the isometries are merely the permutations of the coordinates, that is the elements of, the permutation group of.
Two codes and are said to be equivalent if there exists an isometry of such that. An automorphism of is a Hamming isometry such that. The automorphisms of form a subgroup of called the automorphism group of and we denote it by. Note also that the vector
space can be endowed with a product, so that becomes
a Boolean ring. Furthermore, , for every. The code is said
a constant-weight code (CW-code) if all nonzero codewords have the same weight. The dual of binary
Hamming codes are simplex codes of parameters. simplex codes are constant
weight code (CW-code).
Any permutation of the columns of a k by n binary matrix which maps the rows of into rows of the same matrix, is called an automorphism of the binary matrix [1] . The set of all automorphisms of is a subgroup of the symmetric group and we denote it by. More treatment of linear codes can be found in the book [2] .
Ideally, we would like the rate to be high, in order to be able to send a large number of errors. The
rate of a DW-code approch zero very quickly when the code length increase: as shown
in Figure 1 where and, so.
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.
2. Distinct weight codes
Definition 1 A linear binary code of length is said to be a Distinct Weight Code, (in short: DW-code), if
the weight mapping:, is one to one, that is whenever,.
The simplest example of such codes are the repetition codes. Later we shall give more nontrivial examples. Let a DW-code of length and dimension. Since the number of element of is, then we have. In the sequel we fix our interest to the extreme case, in which we give a construction.
Proposition 2 Let such that. Then every family of words such that is linearly independent.
Proof. Suppose on the contrary that are not linearly independent, then we have a linear combi-
nation, where some is nonzero. Let be the maximal integer such that. Then
, and. Now taking the weights leads to:
a contradiction.
Now we give a construction of a DW-code.
Let be a nonzero integer and. Take the canonical basis of. Put,
then clearly. By the proposition 2, the code-words are linearly independent and
generate a linear code that we denotes by. It also seen that, whenever. This
Proof. Since the cardinal of is, it suffices to show that wt: is onto.
Let, then can be written in the base 2, where. Set,
then.
Up an equivalence we have the following result:
Theorem 4 There exists only one distinct weight -code, moreover such code is Boolean subring of
.
Proof. Let be such a code and take code-words, each has weight. These are linearly independent and form a basis of. Next we show that,. Otherwise, there exists a
least integer such that for some. Since, one have.
Multiplying by yields. Now consider the word,
On the other hand, if we
consider, then. Thus hence
a contradition. This means that, if. Since, and is a
basis of, then is a Boolean ring.
Now we define a linear mapping by. Then,. If
, then. This implies that
is an isometry between and, and by the extension theorem of MacWilliams, see [3] or [4] , there
exists a permutation, such that.
Example 5 and
By using the software Q-extension, see [5] we show, up to equivallence, that among six equivallence classes
the unique DW-code of parametters is the code of generator matrix. It is clear
that it is equivallent to the code of generator matrix. Just swap the second and third
rows and then apply the permutation.
Theorem 6 Let, Diophantine equations for which
, , ,
have a unique solution which is the k-uplet.
Proof. is clearly a solution of the Diophantine equation which satisfies the conditions
(1). Assume that for some and, then where
and. We can assume without loss of generality that
. So by the uniqueness of Development of any integer less than or equal
in binary basis, the equality leads to a contradiction. So the solution
satisfies the conditions (2).
Conversely, Let a solution of the equation satisfying (1)-(2). We can
take, elements of such that and,
, , , , are linearly independent. The
condition (2) means that the code of generator matrix is a dw-code. On after Theorem 1.3, the
condition (1) implies that there exists an invertible by matrix and a permutation matrix
such that where and
is the generator matrix of the code. It is clear that is of the form:
where or 1, and we have. So we have
, and then we have, by the
uniqueness of development of in binary basis. By (1) we have
, then we have:, Kronecker symbol.
Since, we have, and finally we have
.
Remark 7 Without the conditions (1) and (2), Diophantine equations have different solutions. For all
note that there is no DW-self-dual code. Indeed, if not, we will have, wich is impossible.
3. classification and automorphism group of DW-Codes3.1. Automorphism Group: thegeneral case
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 any basis of an DW-code. Then
.
Moreover, if any generator matrix of, then is an automorphism of, if and only if, is an automorphism of the binary matrix.
Proof. Clear.
Proposition 9 The automorphism group of any DW-code is nontrivial of even order.
Proof. Let be a generator matrix of a DW -code. We may suppose that all columns of are
nonzero. The columns of are taken among a set of columns. Suppose that all columns of are distincts, since, then the columns of are the distinct nonzero vectors of and will be the simplex code, which is clearly not DW. This contradiction shows that at least 2 columns of are identical. Now the transposition of these two columns gives an automorphism of.
We deduce that the dual code of a DW-code has a non-trivial automorphism group and has minimum distance.
We consider the general case. The action of automorphism group on the set
of columns of a generator matrix defined by: for all in and
in, splits all the columns of into disjoint orbits. The orbits each formed of a single
column, they are the columns fixed by the group. We set if no orbit is formed of a single
column and then it is clear that since since can not be trivial. The other orbits
are, ,. We set, if and
, , therefore, we have precisely.
Up to equivalence, we can consider that the code $mathcal $ is of generator matrix
rows of, such that.
Since is a DW-code, then for each for each we have
. So we have. We therefore deduce that:,
. So each orbit consists of equal columns.
The following theorem legitimate the idea of giving a definition to the 3-tuple
which we call signature of the DW-code and we denote.
We give here the full classification of such a code in several cases.
Theorem 10 If two DW-codes and are equivalent then they have the same signatures:
.
Proof. Let and two equivalent DW-codes of parameters. So such as.
We have. Let be a generator matrix of the code. Under the action of the
automorphism group we can assume that is of the form where
, and .
So we have and then
which is an orbit of the column under the action of on
the generator matrix of the code. similarly we have
which is an orbit of the column under the action of on the generator matrix. Thus
and have the same number of ponctual orbits, the same number of non-ponctual orbits and the two orbits
and on the one hand and and on the other hand have the same number of columns. we
conclude that the two codes and have the same Signature.
Theorem 25 If is an DW-code with punctual orbits and if the number of non
punctual orbits is greater than the dimension of the DW-code then the code is equivallent to the
DW-code of generator matrix with.
Example 26 The DW-code of generator matrix is equivallent to the code
of generator matrix , , , , , ,
Theorem 27 If is an DW-code with punctual orbits and if the number of non
punctual orbits is lower than the dimension of the DW-code then the code is equivallent to the DW-code of generator matrix
with.
Example 28 The DW-code of generator matrix is equivallent to the code
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 or (and)then, up to equivalence, a generator matrix of the code is of the form where is not an empty submatrix. If the code is self-othogonal then. So
and then we have:
and and then
and then where.
and then
finally and and then
・ If and the code is self-orthogonal if and only if for all.
3.3. Determination of the Automorphism Group
Theorem 30 The automorphism group of a DW-code of signature is isomorphic to
the group direct product.
Proof. Let be a generator matrix of the code. We can assume that is of the form
where, and
.
For, let, let for all. Clearly, the subsets
form a partition of. Now let.
Clearly the are subgroups of and each is isomorphic to and for all
. Since forall, , it follows that the are subgroups of.
Now we are going to show that is the inner direct product
If, and, , then.
Let, then applying this equality to each yields,.
Now let. Since, the are globally invariant under. Let the permutation
defined by, if, and elswhere. Then it is clear that, and this finishes the proof.
Example 31
・ Consider the DW-code of generator matrix. It is equivallente to the code
of generator matrix , , , , ,
and
・ Consider the DW-code of generator matrix. It is equivallente to the code
of generator matrix , , , , and
Acknowledgements
The authors would like to thank the refrees for their helpful suggestions and remarks.
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-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.
