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
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
GG0
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
nnTT
.
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.
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