On the Automorphism Group of Distinct Weight Codes

In this work, we study binary linear distinct weight codes (DW-code). We give a complete classification of [ ]2 , n k -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.


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 [ ] 2 , n k -binary code every k dimensional subspace  of 2 n  .Recall also that the Hamming weight ( ) wt x of vector x is defined to be the number of nonzero components of x .The minimum of weights where 0 x ≠ is the minimal distance d of the code.A Hamming isometry of 2 n  is a linear application x ∈  .It is well known that in binary case, the isometries are merely the permutations of the coordinates, that is the elements of n  , the permutation group of { } 1, 2, , n  .Two codes  and ′  are said to be equivalent if there exists an isometry σ of 2 n  such that ( ) An automorphism of  is a Hamming isometry σ such that ( )  .The automorphisms of  form a subgroup of n  called the automorphism group of  and we denote it by ( ) Aut  .Note also that the vector space n 2  can be endowed with a product ( ) ( ) ( ) Any permutation of the columns of a k by n binary matrix G which maps the rows of G into rows of the same matrix, is called an automorphism of the binary matrix G [1].The set of all automorphisms of G is a subgroup of the symmetric group n S and we denote it by ( ) Aut G .More treatment of linear codes can be found in the book [2].
Ideally, we would like the rate k R n = 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: ( ) ( ) in Figure 1 where k R n = and ( ) 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.

Distinct Weight Codes
Definition 1 A linear binary code  of length n is said to be a Distinct Weight Code, (in short: DW-code), if the weight mapping: , is one to one, that is x y = whenever ( ) ( ) The simplest example of such codes are the repetition codes.Later we shall give more nontrivial examples.Let  a DW-code of length n and dimension k .Since the number of element of  is 2 k , then we have 2 1 k n ≤ + .In the sequel we fix our interest to the extreme case 2 1 ( ) ( ) Let k be a nonzero integer and 2 1 ( )

R r k ≤
where ( ) 0 implies that ( ) ( ) 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 Proof.Since the cardinal of ( ) Up an equivalence we have the following result: Theorem 4 There exists only one distinct weight 2 1, Proof.Let  be such a code and take code-words 1 2 , , , k u u u  , each i u has weight 1 2 i− .These are linearly independent and form a basis of  .Next we show that 0 s r u u = , s r ∀ < .Otherwise, there exists a least integer r such that 0 s r u u ≠ for some s r < .Since . This implies that f is an isometry between  and ( ) , and by the extension theorem of MacWilliams, see [3] or [4], there exists a permutation n σ ∈  , such that ( ) ( ) By using the software Q-extension, see [5] we show, up to equivallence, that among six equivallence classes the unique DW-code 3 7,3,1 is the code of generator matrix 3 0000100 1110010 0001001 . Just swap the second and third rows and then apply the permutation ( )( )( ) have a unique solution which is the k-uplet ( ) ( ) Proof. ( ) 1, 2, 2 , 2 , , 2 k −  is clearly a solution of the Diophantine equation which satisfies the conditions (1).Assume that for some i and I , then 2 2 2 We can assume without loss of generality that . So by the uniqueness of Development of any integer less than or equal 2 1 k − in binary basis, the equality 2 2 2 ∑ leads to a contradiction.So the solution ( ) Conversely, Let ( )  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 k by k matrix ( ) = and a permutation matrix where n S σ ∈ and 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 where , 0 . So we have , and then we have by the uniqueness of development of 2 1 k − in binary basis.By (1) we have and finally we have ( ) ( ) Remark 7 Without the conditions (1) and (2), Diophantine equations have 3 k ≥ note that there is no DW-self-dual code.Indeed, if not, we will have 2 1 2

Automorphism Group: The General 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: Moreover, if G any generator matrix of  , then σ is an automorphism of  , if and only if, σ is an automorphism of the binary matrix G .
Proof.Clear.Proposition 9 The automorphism group of any DW-code is nontrivial of even order.
Proof.Let G be a generator matrix of a DW [ ] , n k -code  .We may suppose that all columns of G are nonzero.The n columns of G are taken among a set of 2 1 k − columns.Suppose that all columns of G are distincts, since 2 1  and  will be the simplex code, which is clearly not DW.This contradiction shows that at least 2 columns of G 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 2 d ⊥ = .We consider the general case 2 1 k n < + .The action of automorphism group ( ) of columns of a generator matrix G defined by: ( ) ( ) each formed of a single column, they are the columns fixed by the group ( ) Aut  .We set 0 f = if no orbit is formed of a single column and then it is clear that 0 2 1 k f ≤ < − since since ( ) Aut  can not be trivial.The ( ) , , , Up to equivalence, we can consider that the code  is of generator matrix Since  is a DW-code, then for each 1, 2, ,  we have ( ) . We therefore deduce that: The following theorem legitimate the idea of giving a definition to the 3-tuple , , , , , r f k t t t  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: ( ) ( ) We have Let G be a generator matrix of the code  .Under the action of the automorphism group

( )
Aut  we can assume that G is of the form ( ) , , , , , , which is an orbit of the column

( )
Aut ′  on the generator matrix G′ .Thus G and G′ have the same number of ponctual the same non-ponctual orbits and the two orbits , n k DW-code without punctual orbits ( ) and if the number of non punctual orbits is equal to the dimension of the DW-code ( ) r k = then the code  is equivallent to a DW- code of generator matrix ( ) 1 1000 0 0 0 01 1 0 0 , , , 000000 001 1 Proof.After a series of permutations and elementary operations on rows of G we can make the first line of the first orbit formed only by ones and all other rows are null O will be formed only by 1 s.And a series 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 1 1000 0 0 0 01 1 0 0 , , , 000000 001 1 It is clear that we have ( ) , n k DW-code admits the system { } , , , k d d d  as orthogonal basis: 2 0 00 00 01 1 . It is equivallente to the code of generator matrix 110000000000000 001111000000000 000000111111111 , n k DW-codes  and ′  without punctual orbits and the number of their orbits is equal to their dimension.Then the codes  and ′  are equivallent if and only if ( ) ( )  . The converse of Theorem 11 is true under an additional condition.
, , , ( as in the last remark ) .If: (1) (1) Proof.Let the application that maps each equivalence class represented by the matrix , , , The converse of this theorem is true under an additional condition.Let  an [ ] 2 , n k of generator matrix , , , f c c c  .So by setting the numbers , , , we have the following result, let  an [ ] n k code of generator matrix G as described in theorem 17 we have: , for all i I ∈ and for all ( ) , n k DW-code such as 2 1 different way to the choice of f fixed columns.For each value of f and for the f s -th choice of f fixed columns we denote by ( ) the number of solutions of the Diophantine equations So we have the following result.Theorem 20 1) The number of equivalence classes of [ ] 2) The number of equivalence classes of [ ] 2 , n k DW-codes with 2 1 , Example 21 By using the result of the last theorem and the Q-extension software, We show that there exist 11,3 DW-code up to equvallence verifying 3 r k = = .Indeed • For 1 f = the set of possible columns taken in the following order are: So the number of DW-codes with 2 1 • For 2 f = the set of possible columns taken in the following order are: So there is no DW-codes with 2 1 • For 3 f = the set of possible columns taken in the following order are: So there is one DW-codes such as 2 1 We deduce that there is only four [ ] 2 11,3 DW-codes, among 98 equivalence classes, satisfying 3 , n k without punctual orbits ( ) and if the number of non punctual orbits is different from the dimension of the DW-code ( ) < and the code  is equivallent to the DW-code 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 1  the DW-code of generator matrix 1 G and 2  the DW-code of generator matrix 2 G such as 1  and 2  are not equivalent and , n k DW-code with f punctual orbits ( ) and if the number of non punctual orbits is greater than the dimension of the DW-code ( ) r k > then the  is equivallent to the DW-code of generator matrix 15,3, 4 DW-code of generator matrix 111000000000100 111111111110010 111110000001001 is equivallent to the code of generator matrix 1100000 1111111 0 0011000 1111111 1 0000111 0000000 0 , n k DW-code with f punctual orbits ( ) and if the number of non punctual orbits is lower than the dimension of the DW-code ( ) r k < then the code C is equivallent to the DW-code of generator matrix 15,3, 4 DW-code of generator matrix 110000000000100 111111110000010 100000001111001 is equivallent to the code of generator matrix 111110000000 100 000001111111 110 000000000000 111 A code which is equivalent to a self-orthogonal code is also self-orthogonal.The property of self-orthogonality is then an invariant of the equivalence of codes.We then have the following points:  Proof.Let G be a generator matrix of the code  .We can assume that G is of the form ( ) Clearly the i G are subgroups of n  and each is isomorphic to

.
Suppose on the contrary that 1 , , k u u  are not linearly independent, then we have a linear combii α is nonzero.Let r be the maximal integer such that Now taking the weights leads to: Figure 1.
one hand and i O and i O′ on the other hand have the same number of columns.we conclude that the two codes C and C′ have the same Signature. of the first row of the generator matrix are zero.Otherwise the first line of another orbit s t

kG
is a generator matrix of a DW-code without punctual orbits ( ) of orbits is equal to the dimension of this DW-code for reasoning by induction.We obtain a generator matrix of an equivalent code and

ω
 the weight of the sum of all the jth rows where − = and we have: So there is no DW-codes avec2 1k n < + , 3 f = and k r= since set of possible columns taken in the following order are: r k ≠ )then, up to equivalence, a generator matrix of the code is of the form [ ] G TD = where D is not an empty submatrix.If the code is self-othogonal then T 0 GG = .So [ ][ ] T 0 TD TD = and then we have: The code  is said a constant-weight code (CW-code) if all nonzero codewords have the same weight.The dual of binary Hamming codes 2, n x y ∈  . 