### Paper Menu >>

### Journal Menu >>

Intelligent Information Management, 2012, 4, 309-310 http://dx.doi.org/10.4236/iim.2012.425043 Published Online October 2012 (http://www.SciRP.org/journal/iim) Automorphism of Cyclic Codes Naser Amiri Department of Mathematices, Payame Noor University, Tehran, Iran Email: n_amiri@pnu.ac.ir Received July 24, 2012; revised August 27, 2012; accepted September 14, 2012 ABSTRACT We investigate how the code automorphism group can be used to study such combinatorial object as codes. Consider GF(qn) as a vector over GF(q). For any k = 0, 1, 2, 3, ···, n. Which GF(qn) exactly one subspace C of dimension k and which is invariant under the automorphism. Keywords: Code; Cyclic Code; Automorphism 1. Introduction There are many kind of automorphism group in mathe- matics. Among them such automorphism group as a code automorphism, (see [1-3]) design automorphism is im- portant to combinatories. These automorphism groups play a key role in determining the corresponding struc- ture, and provide a playground to study elementary alge- bra. 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. 2. Codes and Code Automorphism Group Let F be a finite field. Any subset C of Fn is called an q-array code where F GF q n CGFq . If C is a subset of Fn, then C is called a linear code. In this section we intro- duce basic definition related to code automorphism group, and introduce some computation to find the weight dis- tribution of a code using its automorphism group. Definition 2.1. Let C be a binary code of length n. The binary code of length of n + 1 obtained from C by adding check bit is called the extended code of C. The permuta- tion of coordinate place which send C into itself from the code automorphism group of C denoted by Aut(C). Two binary codes C1 and C2 are equivalent if there is a per- mutation of coordinate place which sends C1 onto C2. If is a none binary code, Aut(C) con- sists of all monomial matrix A over GF(q) such that C for all C C . Note that if two binary codes C1 and C2 are equivalent, then Aut(C1) and Aut(C2) are iso- morphic. But the converse may not always hold. Let C be a binary code and H be a subgroup of Aut(C). For a codeword , the number of 1 in the coordinate place of v is the weight of v and denoted by wt . Usually Ni denotes the number of codewords in C of weight i and Ni(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 sub- group of Aut(C). Then (mode O(H)). NNH C ii Proof. The codewords of weight i can be divided into two classes those fixed by some element of H. If is not fixed by any element of H then the O(H) codeword g for g in H must be distinct. Then Ni – Ni (H) is multiple of O(H). Recall that an action of a group on a set X is the func- tion : f GX X , such that f gx is denoted by gx and with the following properties. 121 2 g gxggx 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 Gggxx 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 1,2,3,, X n ,,,C C. A subset 12 K YC 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 ci. A strong generators for G on X relative to the ordered coordinate bases 12 ,,, K CC C is a generating set S for G such that 12 j cc is generating by c G12 ,,, j cc c SG , for j = 1, 2, 3, ···, t. j Let 1 j j GG0 GG cc, 1: and j jGG 1, 2,,jk , for . Note that Aut J OC , when C opyright © 2012 SciRes. IIM N. AMIRI 310 AutGC nnTT . Definition 2.4. A Hadamard matrix H of order n is an matrix of 1 and –1 such that n H HHHnI. Two Hadamard matrix are equivalent if one can obtained from other by permutating rows and column and multi- plying rows and columns by −1. 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 AH and AN(H) be the matrices obtained from H and NH respectively by replacing 1 with 0. Let CH and N(H) be the binary codes generated by the rows of AH and AN(H) respectively. Theorem 2.5. Let H be a 4m × 4m normalized Ha- damard matrix and m be an even integer. Then CH is the extended code of CN(H). Proof. Let E be the extended code of CN(H). Note that AN(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 AN(H) is also m – 1, since AN(H) is an incidence ma- trix and = b. Also the sum of all rows of AN(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 AH is an element of E. Hence CH is the extended code of CN(H). 3. Cyclic Codes Let V be the n-dimensional vector space over 2n the finite field m p Z or m GF p and the endo- morphism which causes a cyclic shift of coordinate ac- cording to a given bases 01 ,, ,ee e a m p 1n. 01 1102n . We consider the n-dimensional linear space generated by the elements of as a vector space over GF and take the bases ,,,, nn aa a ,,aa mn GF p a normal bases 01 ,, n eaeae m 1 1 , na . Where p aa , . mn aGFp aV a Definition 3.1. A k-dimensional linear subspace C of V is called cyclic code if for all as a in C then in C. We will recall that 1 mn x x VGFp , and C is a cyclic code iff C is an ideal of 1 mn GF pxx. The generator polynomial g of C is the monic polynomial of lowest degree in the ideal, g is a divisor of xn – 1 and dimension(C) = –degngx , and deg 1 2 ,, ,, n gxgxg x is a bases of C. ord m p n is the order of pm in Zn. And Recall that n Qx the nth cyclothymic polynomial. We have 1 nn x Qx . Proposition 3.2. Let 1 mn VGFpx x a npb , and with ,1pb . Then we have: 1) If b = 1 or 2 a bp orb m pb is a prime and b, where is the Uler function, then for any exists exactly one cyclic code of dimension k. 1, 2,,kn 0,1,, a tp Proof. If b = 1, then for . Then only cy- 1 1t n clic code of dimension t is x Qx orb m bpb 12 ,0,, b tt p . If b = pa + 2 where b is prime with , then for the only cyclic code of dimension 12 1 b tp t is 12 12 1 n tt x Qx Qx . a nbp pGF be a vector space Theorem 3.3. Let over n GF p0,, a kp n GF p 2 n bp . Then for any there is ex- actly one cyclic code of dimension k which is invariant under the automorphism if and only if b = 1 or . Proof. It is clearly. 4. Acknowledgements I would like to thank the referees for their valuable common and suggestion leading to this improved version of the paper. REFERENCES [1] A. A. Andrad and R. Palazzo Jr., “Constraction and De- coding of BCH Codes over Finite Commutative Ring,” Linear Algebra and Its Applications, Vol. 286, 1999, pp. 69-85. [2] A. A. Andrad and R. Palazzo Jr., “On Coding Collineat- ing Graphs Of Symmetric Block Design,” Journal of Combinatorial Theory, Vol. 11, No. 3, 1971, pp. 272-281. [3] R. C. Bose and D. K. Ray-Chaudhuri, “On a Class of Error-Correcting Binary Group Codes,” Information and Control, Vol. 3, No. 1, 1960, pp. 68-79. Copyright © 2012 SciRes. IIM |