This document discusses irreducible cyclic codes of length n over finite fields. It begins by reviewing some preliminary concepts regarding characters of finite abelian groups. It then provides theorems that give the irreducible factorization of polynomials of the form x^n-1 over finite fields under certain conditions on n and the field size. Finally, it determines the primitive idempotents in the ring Fq[x]/(x^n-1) and obtains the check polynomials, dimensions, and minimum Hamming distances of the irreducible cyclic codes generated by these idempotents. The key results are the theorems providing the irreducible factorizations of x^n-1 and determining the corresponding primitive idempotents.

1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 10 || December. 2016 || PP-44-68
www.ijmsi.org 44 | Page
The Minimum Hamming Distances of the Irreducible Cyclic
Codes of Length
Sunil Kumar1
, Pankaj2
and Manju Pruthi3
1,2,3
Department of Mathematics, Indira Gandhi University, Meerpur (Rewari)-122502, Haryana, India
Abstract: Let be a finite field with elements and where are positive
integers and are distinct odd primes and 1. In this paper, we study the irreducible
factorization of over and all primitive idempotents in the ring
Moreover, we obtain the dimensions and the minimum Hamming distances of all
irreducible cyclic codes of length over
Keywords: Irreducible factorization, Primitive idempotent, Irreducible cyclic code
2010 Mathematics Subject Classification: 11Txx, 11T71, 68P30, 94Bxx, 94B15
I. INTRODUCTION
Let be a finite field with elements, where and is a prime. A code over the finite field
is called a cyclic if implies Cyclic codes of length
over can be views as ideals in the ring n [x] A cyclic code is called minimal if the
corresponding ideal is minimal. In fact, many well known codes, such as BCH and Hamming codes are cyclic
codes, and many other famous codes can also be constructed from cyclic codes, for example the Kerdock codes
and Golay codes. Cyclic codes also have practical applications, as they can be efficiently encoded by shift
registers. It is true that every cyclic code turns out to be a direct sum of some minimal cyclic codes.
Recently, a lot of papers investigate the minimal cyclic codes, see e.g. [1-6,9,18-21]. It is well known
that minimal cyclic codes and primitive idempotents have a one-to-one correspondence, as every minimal cyclic
code can be generated by exactly one primitive idempotent, so it is useful to determine the primitive
idempotents in [x] In [18], Arora and Pruthi obtained the minimal cyclic codes of length in
over where is an odd prime and they also got explicit expression for the primitive
idempotents, generator polynomials, minimum, distance and dimension of these codes in In
[19], Pruthi and Arora also studied minimal cyclic codes of length over where is a primitive root
modulo they described the -cyclotomic cosets modulo and the primitive idempotents in
Batra and Arora alos obtained the the primitive idempotents in with
in [2] and [5]. In [6], Batra and Arora also obtained the the primitive idempotents in
where is odd and . Sharma et al. obtained all the primitive idempotents
and irreducible cyclic codes in where In [4], Bakshi and Raka showed
all the primitive idempotents in where are distinct odd primes, is a common
primitive root modulo and and In [23], Singh and Pruthi presented explicit
expressions for all the primitive idempotents in the ring where
are distinct odd primes, and and and
In [8], Chen et al. recursively gave minimal cyclic codes of length over where
is an odd prime and In [13], Fengwei Li, Qin Yue and Chengju Li gave primitive idempotents in the
ring where are distinct odd primes and In [12], Fen Li
and Xiwang Cao gave primitive idempotents and minimum hamming distance of all irreducible cyclic codes of
length over where are positive integers and are distinct odd primes and

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In this paper, we shall generalize the results of [12]. Here we shall give all irreducible factors of
over and all primitive idempotents in the ring In Section 2,
we recall some preliminary concept about characters. In Section 3, we focus on factoring polynomials of
over In Section 4, the primitive idempotents in are given. In
Section 5, the check polynomials, dimensions and minimum Hamming distances of the irreducible cyclic codes
of length generated by these primitive idempotents are obtained.
II. CHARACTERS
Let G be a finite abelian group of order n, the finite field of order and A group
homomorphism from G into is called a character of G. It is obvious that the set of all the characters of G
form a finite abelian group under the multiplication of characters.
Lemma 2.1: (The orthogonality relations for characters) (see [14, p. 189])
(1) Let and be characters of G. Then
(2) Furthermore, if and are elements of G, then
The number of characters of a finite abelian group G is equal to . Let G and
Then
is called character matrix of G and
III. FACTORING POLYNOMIALS OF
Let be a finite field with elements, where and is a prime. Every cyclic code of length n
over a finite field is identified with exactly one ideal of the quotient algebra [x] . This is one of
the principal reasons why factoring is so useful. Recently, Chen et al. [7] gave the irreducible
factorization of over where and The irreducible factors of
and over were explicitly obtained in [8] and [9], respectively, where is a prime divisor
of .
In this section, first we will give all irreducible divisors of over with are
positive integers where are distinct odd primes and and then generalize the results for
As usual, we adopt the notations: means that the integer divides and for a prime integer ,
means that but There is a criterion on irreducible non-linear binomials over which was
given by Serret in 1866 (see [14, Theorem3.75] or [24, Theorem10.7]).
Lemma 3.1: Assume that For any with ord the binomial is irreducible over if
and only if both the following two conditions are satisfied:
(i) Every prime divisor of n divides but does not divide ;
(ii) If , then .
Let be a finite field of odd order We denote all the non-zero elements of by i.e., the multiplicative
group of Let ξ be a primitive root of .

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In this paper, we always assume that and , where are
distinct odd primes and , are positive integers. Then , where
In the following, we investigate the irreducible factorization of over By the Division
Algorithm, we have that
For convenience, we will assume that and in this paper. We will denote by a
primitive e-th root of unity over .
When and there is an irreducible factorization over
Theorem 3.2: If and then there is a factorization over
Moreover, suppose that gcd =1.
(1) If then the polynomial is irreducible over
(2) If then the irreducible factorization of over is given as follows:
(3) If then the irreducible factorization of over is given as follows:
Proof: Since and we have
Suppose that gcd =1.
If then so by By Lemma 3.1, polynomial
is irreducible over
If then there is a factorization over
Since and and where
Hence
By and lemma 3.1, we have, is irreducible over
If then the irreducible factorization of over is given as follows:

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By symmetry, we get the following theorem:
Theorem 3.3: If and then there is a factorization over
Moreover, suppose that gcd =1.
(1) If then the polynomial is irreducible over
(2) If then the irreducible factorization of over is given as follows:
(3) If then the irreducible factorization of over is given as follows:
Similarly we can factorize when
(i)
(ii)
Theorem 3.4: If and then there is a factorization over
where we denote as simple.
Moreover, suppose that gcd =1.
When then the polynomial is irreducible over
When we consider the irreducible decomposition of over in the following four
cases:
(1) If and then there the irreducible factorization of over is given as
follows:
(2) If and then the irreducible factorization of over is given as follows:
(3) If and then the irreducible factorization of over is given as follows:
(4) If and then is the irreducible factorization of over is given as follows:

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Proof: When , then and so By Lemma 3.1,
the polynomial is irreducible over
When we consider the irreducible decomposition of over in the following four
cases:
(1) If and then there is a factorization over
Suppose that and then and
Since so and
Hence by Lemma 3.1, the polynomial is irreducible over
(2) If and then the irreducible factorization of over is given as follows:
(3) If and then the irreducible factorization of over is given as follows:
(4) If and then the irreducible factorization of over is given as follows:
This completes the proof of the theorem.
Similarly we can factorize when
(i)
(ii)
(iii)
(iv)
(v)
Theorem 3.5: If and then there is a factorization over
where we denote as simple.
Moreover, suppose that gcd .
When , then the polynomial is irreducible over
When we consider the irreducible decomposition of over in the
following eight cases:
(1) If and then the irreducible factorization of over is
given as follows:

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(2) If and then the irreducible factorization of over is given
as follows:
(3) If and then the irreducible factorization of over is given
as follows:
(4) If and then the irreducible factorization of over is given
as follows:
(5) If and then the irreducible factorization of over is given as
follows:
(6) If and then the irreducible factorization of over is given as
follows:
(7) If and then the irreducible factorization of over is given as
follows:
(8) If and then the irreducible factorization of over is given as follows:
Proof: Proof is similar as that of theorem 3.4.
Similarly we can factorize when
(i)
(ii)
(iii)
Theorem 3.6: If and then there is a factorization over
where we denote as simple.
Moreover, suppose that gcd .
When , then the polynomial is irreducible over
When we consider the irreducible decomposition of over in the
following sixteen cases:

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(1) If and then the irreducible factorization of
over is given as follows:
(2) If and then the irreducible factorization of
over is given as follows:
(3) If and then the irreducible factorization of
over is given as follows:
(4) If and then the irreducible factorization of
over is given as follows:
(5) If and then the irreducible factorization of
over is given as follows:
(6) If and then the irreducible factorization of
over is given as follows:
(7) If and then the irreducible factorization of
over is given as follows:
(8) If and then the irreducible factorization of
over is given as follows:
(9) If and then the irreducible factorization of
over is given as follows:
(10) If and then the irreducible factorization of
over is given as follows:

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(11) If and then the irreducible factorization of
over is given as follows:
(12) If and then the irreducible factorization of
over is given as follows:
(13) If and then the irreducible factorization of
over is given as follows:
(14) If and then the irreducible factorization of
over is given as follows:
(15) If and then the irreducible factorization of
over is given as follows:
(16) If and then the irreducible factorization of
over is given as follows:
Proof: Proof is similar as that of theorem 3.4.
Now we will give all irreducible divisors of over with where
are positive integers and are distinct odd primes and
By the Division Algorithm, we have that
When there is the irreducible factorization over
Now we have the following theorems as generalization of above results:
Theorem 3.7: If and then there is a factorization over

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where we denote as simple.
Moreover, suppose that gcd =1.
When for all then the polynomial is irreducible over When for all we
consider the irreducible decomposition of over in the following cases:
(1) If for all then the irreducible factorization of over is given as follows:
(2) If and then the irreducible factorization of
over is given as follows:
(3) If for all then the irreducible factorization of over is given as follows:
Proof: When for all then and By Lemma 3.1, the
polynomial is irreducible over
Suppose that gcd =1.
When for all we consider the irreducible decomposition of over in the following
cases:
(1) If for all then there is a factorization over
Suppose that then
Since , and so , Hence by
Lemma 3.1, the polynomial is irreducible over
(2) If and then the irreducible factorization of
over is given as follows:

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(3) If for all then the irreducible factorization of over is given as follows:
Theorem 3.8: If and then there is a factorization over
where we denote as simple.
Moreover, suppose that gcd =1.
When for all then the polynomial is irreducible over
When for all we consider the irreducible decomposition of over in the
following cases:
(1) If for all then the irreducible factorization of over is given
as follows:
(2) If and then the irreducible factorization of
over is given as follows:
(3.37)
(3) If and then the irreducible factorization
of over is given as follows:
(3.38)
(4) If and for all then the irreducible factorization of over is given as
follows:

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Proof: Proof is similar as that of theorem 3.7.
IV. PRIMITIVE IDEMPOTENTS IN
Let G = be a cyclic group of order n and a generator of G. Suppose that then
are group homomorphisms, where is a primitive n-th root of unity in They are all characters of G, i.e.
.
Lemma 4.1 [12]: Let G = be a cyclic group of order n and a finite field of order Suppose that
then there are primitive idempotents in
In the following, let and we always assume that and
where are distinct odd primes and
If i.e. then the
primitive idempotents in are obtained (see lemma 4.1). Hence first we shall determine the
primitive idempotents in for under the conditions
and and then determine the primitive idempotents for
general case.
Theorem 4.2: In Theorem 3.2, suppose that gcd . Then all the primitive idempotents in
are given as follows:
(1) If , then the irreducible polynomial over corresponds to the primitive
idempotent:
(2) If then each irreducible polynomial over for
0 corresponds to the primitive idempotent:
(3) If then each irreducible polynomial over for 0
corresponds to the primitive idempotent:
Proof: Since is a primitive -th root of unity in there is a decomposition over
Suppose that gcd =1, then we divide into three cases:

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(1) If , then the polynomial is irreducible over so
contains the only primitive idempotent 1. For convenience, let
and then by the Chinese Remainder Theorem, we have the natural -algebra
isomorphism:
(4.1)
where is of order and
By Lemma 4.1, the primitive idempotent corresponding to the irreducible polynomial
over is
(2) If , then by Theorem 3.2, there is an irreducible factorization over
First, we find all the primitive idempotents in the ring It is clear that the prinicipal
ideal In (4.1), let and be replaced by
Similarly, each primitive idempotent of corresponding to each
irreducible polynomial over for 0 is
Second, we lift each primitive idempotent for 0 in to a primitive
idempotent in . In (4.1), is a primitive idempotent in
Suppose that
By Lemma 4.1, we have
Hence each primitive idempotent in corresponding to each irreducible polynomial
over is
(3) If , then similarly, we know all primitive idempotents in
Hence each primitive idempotent in corresponding to each irreducible
polynomial over is

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Remark 4.3: Suppose that then it is just [9, Theorem 3.4].
By symmetry, we get the following theorem:
Theorem 4.4: In Theorem 3.3, suppose that gcd =1. Then all the primitive idempotents in
are given as follows:
(1) If , then the irreducible polynomial over corresponds to the primitive
idempotent:
(2) If then each irreducible polynomial over for
0 corresponds to the primitive idempotent:
(3) If then each irreducible polynomial over for 0
corresponds to the primitive idempotent:
Similarly, we can obtain primitive idempotents in and
If and then there is a factorization in :
where and is a primitive -th root of unity over
Theorem 4.5: In Theorem 3.4, suppose that gcd =1. Then all the primitive idempotents in
are given as follows:
When , each irreducible polynomial over corresponds to the primitive idempotent:
When we divide into four cases:
(1) If and then each irreducible polynomial over for
corresponds to the primitive idempotent:
(2) If and then each irreducible polynomial over for
corresponds to the primitive idempotent:

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(3) If and then each irreducible polynomial over for
corresponds to the primitive idempotent:
(4) If and then each irreducible polynomial over for
corresponds to the primitive idempotent:
Proof: By (4.2), suppose that gcd =1.
When , then the polynomial is irreducible over so
contains the only primitive idempotent 1. For convenience, let and
then by the Chinese Remainder Theorem, we have the natural -algebra isomorphism:
(4.3)
where is of order and
By Lemma 4.1, the primitive idempotent corresponding to the irreducible polynomial over is:
When we divide into four cases:
(1) If and then by Theorem 3.4, there is an irreducible factorization over
First we find all the primitive idempotents in the ring It is clear that the prinicipal ideal
In (4.3), let and be replaced by
Similarly, each primitive idempotent of corresponding to each irreducible polynomial
over for 0 is
Second, we lift each primitive idempotent for 0 in to a primitive
idempotent in . In (4.3), is a primitive idempotent in
Suppose that
. By Lemma 4.1, we have
Hence each primitive idempotent in corresponding to each irreducible
polynomial over is

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(2) If and then by Theorem 3.4, there is an irreducible factorization over
Similarly, we know all primitive idempotents in
Hence each primitive idempotent in corresponding to each irreducible
polynomial over is
(3) If and then by Theorem 3.4, there is an irreducible factorization over
Similarly, we know all primitive idempotents in
Hence each primitive idempotent in corresponding to each irreducible
polynomial over is
(4) If and then by Theorem 3.4, there is an irreducible factorization over
Similarly, we know all primitive idempotents in
Hence each primitive idempotent in corresponding to each irreducible
polynomial over is
Similarly we can find primitive idempotents in in when
(i)
(ii)
(iii)

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(iv)
(v)
By the same method, we get the following theorems:
Theorem 4.6: In Theorem 3.5, suppose that gcd =1 and is a
primitive -th root of unity over
Then all the primitive idempotents in are given as follows:
When , each irreducible polynomial over corresponds to the primitive
idempotent:
When we divide into eight cases:
(1) If and then each irreducible polynomial
over for corresponds to the primitive idempotent:
(2) If and then each irreducible polynomial
over for corresponds to the primitive idempotent:
(3) If and then each irreducible polynomial
over for corresponds to the primitive idempotent:
(4) If and then each irreducible polynomial
over for corresponds to the primitive idempotent:
(5) If and then each irreducible polynomial
over for corresponds to the primitive idempotent:
(6) If and then each irreducible polynomial
over for corresponds to the primitive idempotent:

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(7) If and then each irreducible polynomial
over for corresponds to the primitive idempotent:
(8) If and then each irreducible polynomial
over for corresponds to the primitive idempotent:
Theorem 4.7: In Theorem 3.6, suppose that gcd and
is a primitive -th root of unity over
Then all the primitive idempotents in are given as follows:
When each irreducible polynomial over corresponds to the
primitive idempotent:
When we divide into sixteen cases:
(1) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(2) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(3) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:

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(4) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(5) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(6) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(7) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(8) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(9) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:

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(10) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(11) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(12) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(13) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(14) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
(15) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:

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(16) If and then each irreducible polynomial
over for corresponds to the primitive
idempotent:
Theorem 4.8: In Theorem 3.7, suppose that gcd =1. Then all the primitive
idempotents in are given as follows:
When for all each irreducible polynomial over corresponds to the primitive
idempotent:
When for all we divide into the following cases:
(1) If for all then each irreducible polynomial over for
0 corresponds to the primitive idempotent:
.
(2) If and then each irreducible polynomial
over for
corresponds to the primitive idempotent:
.
(3) If for all then each irreducible polynomial over for
corresponds to the primitive idempotent:

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.
Theorem 4.9: In Theorem 3.8, suppose that gcd =1. Then all the primitive
idempotents in are given as follows:
When for all then each irreducible polynomial over corresponds to the
primitive idempotent:
When for all we divide into the following cases:
(1) If for all then each irreducible polynomial
over for 0 corresponds to the primitive idempotent:
.
(2) If and then each irreducible polynomial
over for
corresponds to the primitive idempotent:
.
(3) If and then each irreducible
polynomial over for
corresponds to the primitive idempotent:

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.
(4) If for all then each irreducible polynomial over for
corresponds to the primitive idempotent:
.
V. THE MINIMUM HAMMING DISTANCES OF THE IRREDUCIBLE CYCLIC CODES
In this section, first we give the check polynomials, minimum Hamming distances and the dimensions
of the irreducible cyclic codes generated by the primitive idempotents in for
and then give generalized result. We indicate that the parameters of the irreducible cyclic codes in
also can be easily obtained based on the primitive idempotents in
Let C denote an irreducible cyclic code of length generated by a primitive idempotent
whose check polynomial is an irreducible divisor of . It is clear that
C where is called the generator polynomial of the
irreducible cyclic code C.
Theorem 5.1: In Theorem 4.2, suppose that gcd
(1) If then Ci is an cyclic code whose check
polynomial is
(2) If Ci,j 0 is an cyclic
code whose check polynomial is
(3) If Ci,j 0 is an cyclic code
whose check polynomial is
Proof: If then by the construction of we have a ring isomorphism:
Ci
where Hence is the check polynomial of Ci. On the other
hand, if then by division algorithm where
Then we have

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Recall that
The degree of each item about differs at least and If then we get the
Hamming weight of in is just
In fact, the equality holds if and only if Thus we have (Ci)
If then by the construction of we have a ring isomorphism:
Ci,j
Hence is the check polynomial of Ci,j.
Recall that
Similarly, we have (Ci,j)
We can also get the proof in the case
By the same method, we can get the following theorems:
Theorem 5.2: In Theorem 4.4, suppose that gcd =1,
(1) If then Ci is an cyclic code whose check polynomial
is
(2) If Ci,j 0 is an cyclic code
whose check polynomial is
(3) If Ci,j 0 is an cyclic code whose
check polynomial is
Theorem 5.3: In Theorem 4.5, suppose that gcd
When , then Ci is an cyclic code whose check
polynomial is
When we divide into four cases:
(1) If and then Ci,j 0 is an
cyclic code whose check polynomial is
(2) If and then Ci,j 0 is an
cyclic code whose check polynomial is
(3) If and then Ci,j 0 is an
cyclic code whose check polynomial is
(4) If and then Ci,j 0 is an
cyclic code whose check polynomial is

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Theorem 5.4: In Theorem 4.6, suppose that gcd
When , then Ci is an cyclic code
whose check polynomial is
When we divide into eight cases:
(1) If and then Ci 0 is an
cyclic code whose check polynomial is
(2) If and then Ci,j 0 is an
cyclic code whose check polynomial is
(3) If and then Ci,j 0 is an
cyclic code whose check polynomial is
(4) If and then Ci,j 0 is an
cyclic code whose check polynomial is
(5) If and then Ci,j 0 is an
cyclic code whose check polynomial is
(6) If and then Ci,j 0 is an
cyclic code whose check polynomial is
(7) If and then Ci,j 0 is an
cyclic code whose check polynomial is
(8) If and then Ci,j 0 is an
cyclic code whose check polynomial is
Theorem 5.5: In Theorem 4.7, suppose that gcd
When , then Ci is an
cyclic code whose check polynomial is
When we divide into sixteen cases:
(1) If and then Ci is an
cyclic code whose check
polynomial is
(2) If and then Ci is an
cyclic code whose check polynomial is

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(3) If and then Ci is an
cyclic code whose check polynomial is
(4) If and then Ci is an
cyclic code whose check polynomial is
(5) If and then Ci is an
cyclic code whose check polynomial is
(6) If and then Ci is an
cyclic code whose check polynomial is
(7) If and then Ci is an
cyclic code whose check polynomial is
(8) If and then Ci is an
cyclic code whose check polynomial is
(9) If and then Ci is an
cyclic code whose check polynomial is
(10) If and then Ci is an
cyclic code whose check polynomial is
(11) If and then Ci is an
cyclic code whose check polynomial is
(12) If and then Ci is an
cyclic code whose check polynomial is
(13) If and then Ci is an
cyclic code whose check polynomial is
(14) If and then Ci is an
cyclic code whose check polynomial is
(15) If and then Ci is an
cyclic code whose check polynomial is

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(16) If and then Ci is an
cyclic code whose check polynomial is
Now we give the following theorems as generalization of above results:
Theorem 5.6: In Theorem 4.8, suppose that gcd
When for all then Ci is an
cyclic code whose check polynomial is
When for all we divide into the following cases:
(1) If for all then Ci 0 is an
cyclic code whose check polynomial is
(2) If and then Ci
is an
cyclic code whose check polynomial is
(3) If for all then Ci is an
cyclic code whose check polynomial is
.
Theorem 5.7: In Theorem 4.9, suppose that gcd
When for all then Ci is an
cyclic code whose check polynomial is
When for all we divide into the following cases:
(1) If for all then Ci 0 is an
cyclic code whose check
polynomial is
(2) If and then Ci
is an
cyclic code whose check polynomial is

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(3) If and then Ci
is an
cyclic code whose check polynomial is
(4) If for all then Ci is an
1, cyclic code whose check polynomial is
REFERENCES
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