Automorphism of Cyclic Codes

We investigate how the code automorphism group can be used to study such combinatorial object as codes. Consider GF(q n) as a vector over GF(q). For any k = 0, 1, 2, 3, ···, n. Which GF(q n) exactly one subspace C of dimension k and which is invariant under the automorphism.


Introduction
There are many kind of automorphism group in mathematics.Among them such automorphism group as a code automorphism, (see [1][2][3]) design automorphism is important to combinatories.These automorphism groups play a key role in determining the corresponding structure, and provide a playground to study elementary algebra.In particular, code automorphism group is useful in determining the structure of codes, computing weight distribution, classifying codes, and devising decoding algorithm, and many kinds of code automorphism group algorithms.In this paper, we will investigate how the code automorphism group can be used to study some combinatorial structures.

Codes and Code Automorphism Group
Let F be a finite field.Any subset C of F n is called an q-array code where In this section we introduce basic definition related to code automorphism group, and introduce some computation to find the weight distribution of a code using its automorphism group.Usually N i denotes the number of codewords in C of weight i and N i (H) the number of codewords which are fixed by some element of H. Now, we will investigate a method of using the automorphism of group to find out the weight distribution of a given code C.
Theorem 2.2.Let C be a binary code and H be a subgroup of Aut(C).Then (mode O(H)).

  N N H 
Proof.The codewords of weight i can be divided into two classes those fixed by some element of H.
Recall that an action of a group on a set X is the func- is denoted by gx and with the following properties.
g g x g g x  and (ex = x).If x, y are in X, we say that x ~ y if there is g in G such that y = gx.And if x in X, we defined x is called the isotropy or (stabilizer) subgroup of G, or subgroup fixing by x.
Definition 2.3.Let C be a binary code of length n and G is a subgroup of Aut(C).Then G acts on the coordinate place Y C of X is called a coordinate base for G provided that the identity element fixes all the coordinate places for G provided that the identity element fixes all the coordinate places c i .
A strong generators for G on X relative to the ordered coordinate bases Let H be 4m × 4m Hadamard matrix and N(H) be the matrix obtained from H by deleting the first row and column.Now let A H and A N(H) be the matrices obtained from H and N H respectively by replacing 1 with 0. Let C H and N(H) be the binary codes generated by the rows of A H and A N(H) respectively.
Theorem 2.5.Let H be a 4m × 4m normalized Hadamard matrix and m be an even integer.Then C H is the extended code of C N(H) .
Proof.Let E be the extended code of C N(H) .Note that A N(H) is an incidence matrix of a (4m -1, 2m -1, m -1) design (i.e.4m -1 point a set of b block.Each block has 2m -1 point.Every 2 point lie on exactly m -1 block).
With b = 4m -1 =  .Note that the number of 1 in each row of A N(H) is also m -1, since A N(H) is an incidence matrix and  = b.Also the sum of all rows of A N(H) is an all one vector since m -1 is an odd integer.Thus all one vector of length 4m is in E, and the length 4m vector is also in E. Note that those vector generate E, an each row of A H is an element of E. Hence C H is the extended code of C N(H) .

Cyclic Codes
Let V be the n-dimensional vector space over   .We consider the n-dimensional linear space generated by the elements of as a vector space over GF and take the bases as a in C then in C. We will recall that The generator polynomial g of C is the monic polynomial of lowest degree in the ideal, g is a divisor of x n -1 and dimension , and is the order of p m in Z n .And Recall that   n Q x the nth cyclothymic polynomial.We have  the endomorphism which causes a cyclic shift of coordinate according to a given bases

..
function, then for any exists exactly one cyclic code of dimension k.If b = p a + 2 where b is prime with , then for the only cyclic code of dimension Then for any there is exactly one cyclic code of dimension k which is invariant under the automorphism if and only if b = 1 or  . Proof.It is clearly.
The binary code of length of n + 1 obtained from C by adding check bit is called the extended code of C.
Definition 2.1.Let C be a binary code of length n. wt  .
2.4.A Hadamard matrix H of order n is an matrix of 1 and -1 such that