Transform Domain Characterization of Dual Group Codes of Cyclic Group Codes over Elementary Abelian Groups

The group of characters of an elementary Abelian group m p Z has been used to define duality between its subgroups, which in turn is extended to duality between group codes. The transform domain description of the dual codes of cyclic group codes of length 1 − = m p n over m p Z has been developed in this paper. Several example codes and their duals have been presented also.

Z , which is the additive group of ) ( m p GF (Zain and Rajan, 1995).The dual codes of systematic group codes over finite Abelian groups have been characterized (Zain and Rajan, 1997) in terms of the endomorphism of the Abelian group that defines the group code.
In this paper, using the structural properties of the dual codes of cyclic group codes of length 1 − = m p n over m p Z , their characterization in the transform domain is presented.Several example codes and their duals are presented also.The paper is organized as follows.Section 2 presents the mathematical preliminaries that are relevant to the development of the main results.In section 3 the main theorem that characterizes the codes and their duals in the transform domain is proved.Illustration of the applicability of the main theorem with examples codes and their duals is presented in section 4. Conclusions and suggestions for further work are given in section 5.

Mathematical preliminaries
The minimum mathematical background that is necessary for the development of the main result of the paper is presented in the following two subsections.

Group characters, inner product and dual subgroups
A character of a group G is a homomorphism of G into the group of units of the field of complex numbers * C or group of units of any appropriate field in which there exist an th m − root of unity, where m is the exponent of G. Let Ĝ be the set of characters of G.
Result (1) ( Hungerford, 1989): Under the above isomorphism, the elements of Ĝ can be indexed with the elements of G as follows: Let H be a subgroup of G .Definition (1) ( Hungerford, 1989): where e is the identity element in the group of units.Definition (2) ( Hungerford, 1989): An inner product on G is defined next.The indexing of the characters of G with the elements of G can be done such that the mapping: given by , ( ), , is symmetric in both arguments.This mapping will be called an inner product on G ( Delsarte 1972).

Let
The EADFT transform defined above is invertible (Zain and Rajan, 1995), and its inverse is given below: The following definition is important to the characterization of the class of codes that are cyclic.Definition (4) (Zain and Rajan, 1997): (Invariant Subgroups): Let G be a group and H be a subgroup of ) (G

Aut
. A subgroup of G which is invariant under the action of H on G is called a H -invariant subgroup of G .Definition (5) (Zain and Rajan, 1997) , and j e is the exponent.

Transform Domain Characterization of Cyclic Group Codes
The transform domain characterization of the class of codes that are group codes ( not necessarily linear codes), cyclic and of length 1 − = m p n over the elementary Abelian group m p Z presented in (Zain and Rajan 1995), identifies two cases as shown below: • Case 1, In this class of cyclic group codes, all the transform components are free.
• Case 2, In this class of cyclic group codes, the transform components that are in the same conjugacy class are related.
In this paper, we consider case 1 which is the class of codes of length n that is equal to 1 − m p , in which the transform components that lie in the same conjugacy class are all either free or zeros.

Main theorem
The following theorem gives the transform domain characterization of the dual codes of the cyclic codes, whose transform components that lie in one conjugacy class are all either free or assigned zeros, (those codes covered by case 1).,  , (

Theorem (1):
Since the components of the transform vector Next we compute the inner product This completes the proof.

Illustrations of the theorem
The usefulness of the main theorem is best demonstrated with the following two numerical examples, where a length 3 cyclic group code over the elementary Abelian group with four elements is characterized in the transform domain, then by using the theorem, its dual is also characterized and obtained.

Example (1):
The detailed mapping is given in Table 1: According to the theorem the dual code should have the following characterization for the conjugacy classes: The detailed listing of the code vectors and the transform vectors of the code and its dual is presented in Table 3.

Conclusions
The structural properties of the dual codes of cyclic group codes of length Z is characterized in the transform domain, then its dual code is obtained.
Further work can be done to generalize the main result to other classes of cyclic group codes whose transform characterization contains components that are related (Zain and Rajan, 1997), and to quasi-cyclic codes (Dey and Rajan, 2003).

Z
has been used to define duality between its subgroups, which in turn is extended to duality between group codes.The transform domain description of the dual codes of cyclic group codes of length1.Introduction he transform domain description of cyclic linear codes over) known(Blahut, 1997).The transform domain description of cyclic group codes over an elementary Abelian group .... times) has been developed byZain and Rajan (1997)  ; this class of codes are subgroups of the times), which amounts to relaxing the condition of linearity, T hence yielding a larger class of codes that contains the class of linear codes.The important class of Reedgroup codes over the elementary Abelian group m p n denote the length of the code where n and p are relatively prime.Definition (3)(Zain and Rajan, 1995): The transform vector of , n is the inverse of an automorphism of m p Z defined by: the transform vectors of the dual code d C take values from the following j S

Z
have been used to present and prove a theorem that gives the characterization, in the transform domain, of the dual codes in terms of the dual subgroups of the group

Transform Domain Characterization of Dual Group Codes of Cyclic Group Codes over Elementary Abelian Groups Adnan Abdulla Zain Department
of Electronics and Communications Engineering, Faculty of Engineering, University of Aden, Yemen.Email: adnan_zain2003@yahoo.com.
ABSTRACT:The group of characters of an elementary Abelian group

Table 2 .
Subgroups and their Duals.The transform vectors for the code are listed in the first column where the characterization for the conjugacy classes is as follows:

Table 3 .
Code vectors and their transforms.