Generating and cyclonomic polynomials of constacyclic codes

Tilburg University

Generating and cyclonomic polynomials of constacyclic codes

van Zanten, A.J.

Publication date: 2014

Document Version Peer reviewed version

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Citation for published version (APA):

van Zanten, A. J. (2014). Generating and cyclonomic polynomials of constacyclic codes. Tilburg University.

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1

Generating and Cyclonomic Polynomials

of

Constacyclic Codes

TiCC, Tilburg University

2

Abstract

In this report we investigate the idempotent generating polynomials of



-constacyclic codes of length n over

GF q

( )

, with

( , ) 1

n q



and





GF q

( )*

, which are defined by factor polynomials of

n

x





. In particular we study the relationship between such polynomials and the idempotent generating polynomials of cyclic codes defined by factors of

x

e



1

, where e stands for the order of

.

n

3

Contents

1. Definition and generators of constacyclic codes p. 4

2. The order of the polynomial

x

n





p. 5

3. Idempotent generators of constacyclic codes p. 8

4. Examples p. 9

5. A relation between idempotent generators of constacyclic and cyclic codes p. 12

6. More examples p. 13

7. Cyclonomic polynomials for constacyclic codes p. 15

8. Factorizations of

x

N



1

into binomials p. 19

9. Computation of idempotents of constacyclic codes from those of cyclic codes p. 21

10. Suggestions for further study p. 26

References p. 27

4

1. Definition and generators of constacyclic codes In this report we study



-constacyclic codes, i.e. codes generated by polynomials which divide

n

x





,





GF q

( )*

. These codes were first introduced in [1]. For more information and for more information we refer to [7,8] and to the lists of references in these publications. In particular we investigate the idempotent generating polynomials (idempotent generators) of such codes. Firstly, we state some properties of idempotent generators which are well known in the case

1



 (cyclic codes), but which also hold for



1. Let n _s( )( ) s S

x



P  x



 



be the decomposition of

x

n





into monic irreducible polynomials over

GF q

( )

. In the next we shall write

P x

_s

( )

instead of P_s( ) ( )x , if no confusion will arise. If f x_s( ) : ( xn



) /P x_s( ), sS, then the code



f x

s

( )



is called a minimal or irreducible



-constacyclic code. In algebraic terms, such a code is minimal ideal in the ring

R

_{n q}_. of polynomials modulo

x

n





over

GF q

( )

. The code



P x

_s

( )



is called a maximal



-constacyclic code. Generally, we shall denote an idempotent generator of a



-constacyclic code



g x

( )



by

e x

( )

or, if necessary to emphasize its



-dependency, by

e x

_

( )

. The idempotent generators of the maximal codes



P x

_s

( )



are denoted by



_s

( )

x

, and those of the minimal or irreducible codes



f x

_s

( )



by



_s

( )

x

. The polynomials



_s

( )

x

, sS, are occasionally called primitive idempotent polynomials because of their properties mentioned in the next theorem.

Theorem 1

(i) If C is a



-constacyclic code, its idempotent generating polynomial is uniquely determined. (ii) If

e x

( )

is the idempotent generator of a code C, then

e x

( )

k



e x

( )

for all k1.

(iii) If

C

₁ and

C

₂ are



-constacyclic codes with idempotent generators

e x

₁

( )

and

e x

₂

( )

, then

1 2

C



C

and

C

₁



C

₂ are also



-constacyclic codes with idempotent generators

e x e x

₁

( ) ( )

₂ and

1

( )

2

( )

1

( ) ( )

2

e x



e x



e x e x

, respectively. (iv) Let S be the index set of the irreducible polynomials contained in

x

n





, then _s( ) 1

5

Theorem 2 (i) Let the



- constacyclic code C in

R

_nq_, be generated by some polynomial

g x

( )

and let

h x

( )

be its check polynomial. The polynomial e x( )R_nq_, is the idempotent generator of C if and only if

( )

0

e





if

g

( )





0

and

e

( ) 1





if

h

( )





0

, for all

n

elements



which satisfy



n





in

some appropriate extension field of

GF q

( )

. (ii)

c x

( )



C

if and only if

e x c x

( ) ( )



c x

( )

.

For the proofs we refer again to [4,5,6], where the case



1 is proven. In the next we shall use the notation

GF q

( )( )



for the extension field of

GF q

( )

by an element



which is algebraic over

( )

GF q

. In the special case that





GF q

( )

, the field

GF q

( )( )



coincides with

GF q

( )

. If



is of order

m

over

GF q

( )

, then ∣

GF q

( )( )



∣ =

q

m, as is well known.

2. The order of the polynomial

x

n





In this section we study the order of the polynomial

x

n





, i.e. the least positive integer

e

such that n

x





is a divisor of

x

e



1

in

GF q x

( )[ ]

(cf.[ 3, Ch. 3]).

Theorem 3 Let n be an integer



1

and q a prime power with

( , ) 1

n q



. Let





GF q

( )*

have order k, and let



and



be elements in some extension field F of

GF q

( )

, such that



n





and



is a primitive

th

n

root of unity. Let furthermore

1 ( ) l s s P x 



be the factorization of

x

n





into irreducible

polynomials over

GF q

( )

, and let

e

be its order in

GF q x

( )[ ]

. (i) The zeros of the polynomial

x

n





in F can be written as



i, 0  i n 1,

x

n





is a divisor

of

x

kn



1

in

GF q x

( )[ ]

and ekn. (ii) If the l irreducible factor polynomials

P x

_s

( )

, 1 s l, of

x

n





in

GF q x

( )[ ]

have orders

e

₁,

2

e

, ....,

e

_l, then

e

:



e e

₁

, ,....,

₂

e

_l



.

(iii) The order

e

of

x

n





is equal to the least common multiple of the orders of its zeros. (iv) If there is no integer i,

0 i

 

n

, such that



i



GF q

( )

, then



has order kn in F. Conversely,

if there is an integer

i

,

0 i

 

n

, with



i



GF q

( )

, then there exists a minimal integer d∣n, d n,

such that



d

 



GF q

( )

, and ord

( )





h

implies that the order of



ishd. (v) The order of



is equal to kn if and only if

n

is a divisor of hd.

(vi) If

n

is not a divisor ofhd, then



is a zero of

x

d





which is a factor of

x

n





and which

has order hd kn.

(vii) For

k

 

q

1

, the order of



and of

x

n





is equal to kn (viii) There exists a zero





F

of

x

n





of order

e

, and the zeros of

x

n





can be written as

1

ik





6

Proof (i) Since

(



i n

)



 

n in





and since all



i, 0  i n 1, are different, the first statement follows immediately. Another argument follows from the fact that the equation

x

n

 



0

is equivalent to 1 0. n x



     

  The second statement is a consequence of

(

)

1

i kn kn k













. (ii) This is an immediate consequence of the definition of the notion of order of a polynomial and of the facts that

x

s



1

∣

x

t



1

if and only if

s

∣t and that

₍

_x

e1



_1,

_x

e2

 

₁₎

_x

( ,e e1 2)



₁

_{(cf. [3, Theorem}

3.9 and Corollary 3.7]). (iii) The order of an irreducible polynomial over

GF q

( )

of degree

m

is equal to the order of any of its zeros



in the multiplicative group

F

*

:



GF q

( )( ) (



*



GF q

(

m

) )

* . The result now follows by applying (ii). (iv) Let the order of



in

F

be equal to

e

. Then we have kne, since



kn





k



1

. On the other hand,

e



n

, since for

e



n

we would have



e

 

1

GF q

( )

contradicting the condition on



. Hence, we can write esn t ,

s



1

, 0 t n. It follows that

1





sn t



 

s t in

F

. Therefore,

t s



_





. Because of the condition on



again, this can only be true for t0, and thus



s



1

. So, sk and ekn. We conclude that ekn. Conversely, let



i



GF q

( )

and

0

 

i

n

. It follows that



an bi



GF q

( )

for all integer values

a

and b. Since the greatest common divisor

( , )

n i

is equal to

an bi



for some particular values of

a

and b, we also have

 

:



d



GF q

( )

*, with

d

: ( , )



n i

. Assume that d is minimal with respect to this property. Moreover, dn and dis a divisor of

n

. Suppose



d





(



GF q

( )*)

, and let the order of



in

GF q

( )*

be h. Then, similar to the first part, we have that the order of



in

F

is hd. (v) In general, we know from (iv) that the order of



is equal to hd. We also know from (iv) that

d∣

n

. Since



d generates a group of order h and



n a subgroup of order k, we have that

/ ( , )

n

k

h

h

d



. It follows that knhd if and only if

( , / )

h n d



n d

/

, or equivalently, if and only if /

n d is a divisor of h. (vi) Since



satisfies



d





, it is a zero of

x

d





. According to (iv) the order of this polynomial

is equal to hd kn. So, hd is a real divisor of kn. (vii) In general k∣h. In this case we must have hk, since

k

(:

 

q

1)

is the maximal divisor of

1

q



. So, nd and knhd. The result now follows from (v) and (ii). (viii) Let

F

'

be an extension field of

F

containing all zeros of

x

e



1

. These zeros constitute a

multiplicative group G of order

e

. Let



be a generator of G. Since

x

n





is a divisor of

x

e



1

. there are

n

different elements



j such that



jn





. If

j

₀ is one of these exponents, the

n

different elements



j j0 _{all satisfy}

₍



j j0

₎

n



₁

_{and so they form a subgroup of order}

_n

_{. Hence,}

_n

_{is a}

7

Since the order of

x

n





is

e

, this polynomial has at least one zero



of order

e

. Therefore, we can take

 



in the above lines, which proves the statements in (viii). □ Example 0 Consider the binomial

x

6



2

over

GF

(5)

. So, n6,



2, k 4 and

q



5

. The binomial can be factorized into irreducible polynomials as follows:

x

6

 

2

(

x

2



2)(

x

2

 

x

2)(

x

2

 

x

2)

.

We extend the field

GF

(5)

by taking for



a zero of

x

2



2

x



1

. This yields the powers

2

2



 



,



3

  



2

,



4



2





2

,



5

  



1

,



6



2

,



12

 

1



24



1

. It follows that the order of



in

F

:



GF

(5)( )



is equal to 244.6kn. This demonstrates the first part of Theorem 3 (iv), since the condition that there is no

j

,

0

 

j

6

, such that



j



GF

(5)

, is satisfied. If we take for



a zero of

x

2



2

, we have the



-powers



2

 

2

,



3  



2,



4



2





2

,

5

1



  



,



6



2

,



8



1

. So, the order of this



in

F

is 8. The second part of Theorem 3 (iv) provides us also with this value. Now, there are integers

j

,

0

 

j

6

, such that



j



GF

(5)

. The smallest of these integers is 2, so d 2 and h4, which indeed yields the order 8 for



. As for Theorem 3 (v), n6 is not a divisor of hd4.28 and neither of h d' '2.48, hence

4.6

kn hd, or equivalently kn is not the order of



. The polynomial

x

2



2

has order 8, and so

2

2

x



∣

x

8



1

.

Next, we consider the binomial

x

6



1

, with



 1, k 2, and which has the factorization into irreducible polynomials over

GF

(5)

x

6

  

1 (

x

2)(

x



2)(

x

2



2

x



1)(

x

2



2

x



1)

.

It appears that the zeros of

x

2



2

x



1

have order 12, and hence the polynomials themselves also have order 12. If we take for



the only zero of x2, i.e.



: 2 , we obtain the powers



2

 

1

,

3

2



 

,



4



1

. So, the order of



in

GF

(5)

as well as in

F

:



GF

(5)

is 4 . Since the order of

1

:

d

2

 









is equal to 4, it follows also from the second part of Theorem 3 (iv) that the order of



, and hence of its defining polynomial x2, is equal to hd 4.14.

Similarly, we can derive for the binomial

x

3

  

1 (

x

1)(

x

2

 

x

1)

over

GF

(5)

, that if



is a zero of

x

2

 

x

1

its order in

F

is equal to 6 and



3

 

1

. It follows that d 3 and h2, and hence, by applying Theorem 3 (iv), the order of



and of

x

2

 

x

1

is equal to 6. In this case we have

3.2 2.3

hd  kn. The order of x1 is equal to 2, and so the order of

x

3



1

is equal to

2, 6

6





.

8

x

6

 

1 (

x

2



1)(

x

2



5

x



1)(

x

2



5

x



1)

.

Taking for



a zero of

x

2



5

x



1

yields the following powers in

GF

(11)

:



0



1

,



6

 

1

,

12

1





. So, the order of



in

F

is 12, and hence the order of

x

2



5

x



1

is also 12. Furthermore, since

 



6

 

1

, we have n6 and k2, giving also this order kn2.6 12 , since there is no

j

,

0

 

j

6

, such that



j



GF

(5)

. □

Theorem 4 Let n be an integer



1

and q a prime power with

( , ) 1

n q



. Let





GF q

( )*

have order k, and let



be an element of order kn in some extension field

F

of

GF q

( )

, such that



n





. Then the

following properties hold. (i)

F

contains



( )

n

primitive

n

th roots of unity



and the n zeros of the polynomial

x

n





in F

can be written as



i, 0  i n 1. (ii) If 1 i



, 2 i



, ...., l i



are zeros of the l irreducible polynomials

1( ) i P x , 2( ) i P x , ...., ( ) l i P x which are contained in

x

n





, then 1 2 : .... l i i i

  





has order e in

F

and

1 2 /

: (

....

)

l e n i i i





 



is a primitive th

n

root of unity. (iii) LetG be the multiplicative group of zeros of

x

e



1

in

F

, and let

H

:

 



be the subgroup consisting of the

n

zeros of

x

n



1

. Then the cosets of H in G are

H

_j





j

H

, 0  j k 1, and we can write 1 0 k j j G H    . (iv) The coset

H

_j contains all n zeros in F of the polynomial

x

n





j, for 0  j k 1. Proof (i) We can take for

F

the splitting field of

x

n





which contains all its

n

zeros . Let



be a

primitive element of

F

, defined by a primitive polynomial of degree

m

. So,

F



GF q

(

m

)

,



generates the group

F

*

and the order of



is m

1

q



. Since the order of

x

n





is equal to

e

, it follows that

e

is a divisor of

q

m



1

by Lemma 3.6 in [3]. Hence, there exists a subgroup G of

F

*

of order

e

and such that its elements are the zeros of

x

e



1

. Let



(





r for some r) be a generator of this subgroup G. So,



e



1

and all elements



i, 0 i e, are different. Among the

e

zeros of

1

e

x



there are the

n

zeros



₀,



₁, ….,



_n1 of

x

n





in

F

. The quotients

 

₀

/

_i, 0  i n 1, are all in G, since G is a group. Since they all satisfy

x

n



1

and are pairwise different, they form the complete set of zeros of the polynomial

x

n



1

. Hence, among these quotients there are



( )

n

primitive

n

th roots of unity. Each of these roots generates the same subgroup

H



G

of order

n

, which is the only subgroup of G of order

n

, since G is cyclic. By indexing the zeros



_i in an appropriate way, we can obtain that



_i1

/

 

_i



, 0  i n 2, where



is one of these, a priori

9 (ii) Let 1 i



, 2 i



, ...., l i



be zeros of the l irreducible polynomials

P x

₁

( )

,

P x

₂

( )

, ....,

P x

_l

( )

respectively, contained in

x

n





. Then the product

1 2.... l

i i i

 



has order

e

in

F

and so we can define

1 2

: ....

l

i i i

  





. In the proof of (i) we showed that among the powers



i, 0 i e, there are

( )

n



elements of order

n

. Since e n/ is an integer and since all powers

(



e n i/

)

, 0  i n 1, are

different,



e n/ is one of these primitive

n

th roots. (iii) Since



n





, we have that



H for



1. Assume that



j





i

H

for some

j

with

i

 

j

k

. Then



j i



H

which is false if



1. So, for



1 all cosets

H

j





j

H

,

0  j k 1, are disjunct and we can write

1 0 1 1 0 ( ) .... k k j j G H H H H H          . For



1

we have k1, and hence this relation is also true in that case.

(iv) This follows immediately from the fact that



n





if and only if

(



j n

)





j, 0  j k 1. □

Remarks

(i) The element



in Theorem 4 (ii) is not a zero of

x

n





. (ii) The proof of the first part of Theorem 3 (iv) is based on the proof of Lemma 3.17 in [3], which

deals with a similar property for the order of an arbitrary polynomial

f



GF q x

( )[ ]

,

f

(0)



0

, of positive degree.

(iii) Theorem 4 (i) and its proof is a slight generalization of Theorem 3 (viii). (iv) If we take

x

n



1

as a polynomial in

GF q x

( )[ ]

, we know that at least one of its zeros is a

primitive element of order

n

, while the orders of all other zeros are divisors of

n

. So, the order of

1

n

x



is equal to

n

, which of course is true.

(v) The approach we referred to in the proof of Theorem 3 (i), i.e. transforming the polynomial

x

n





into



n

(

x

n



1)

, does not provide us with a kind of generalized cyclotomic polynomial. As an example we write

x

6

2



6

(( )

x

6

1)



 



. Putting

y

x





and using the factorization of

y

6



1

into cyclotomic polynomials, yields

x

6

  

2

(

x



)(

x





)(

x

2





x





2

)(

x

2





x





2

)

. Since the polynomials in the rhs are



- dependent, there seems no obvious generalization present of the notion of cyclotomic polynomial.

3. Idempotent generators of constacyclic codes In this section we derive a simple general expression for the uniquely determined idempotent

generator of a constacyclic code. This expression generalizes a similar expression for idempotent generators for cyclic codes which was derived in [9] (cf. also [10, 11]). Furthermore, we discuss a

number of examples. Theorem 5

10

of polynomials mod

x

n





over

GF q

( )

. (i) If the polynomial

g x

( )

is a divisor of

x

n





in

GF q x

( )[ ]

, then the idempotent generator of the code



g x

( )



is given by

e x

( )



(

n



)

1

xh x g x

'( ) ( )

, where

h x

'( )

stands for the formal derivative of the polynomial

h x

( ) : (



x

n





) / ( )

g x

in the ring

R

_nq_,. (ii) The idempotent generator of the dual code



g x

( )



┴ is given by

e x

( )

┴ 1

(

n



)



xg x h x

'( ) ( ).



(iii)

e x

( )



e x

( )

┴ =1.

Proof (i) In general we have that if

a x g x

( ) ( )



b x h x

( ) ( ) 1



in

R

_nq_,, then the polynomial

( ) :

( ) ( )

e x



a x g x

satisfies

e x

( )

2



e x

( )

and hence

e x

( )

is the idempotent generator of the code

( )

g x





. Now, we start by taking derivatives of both sides of the relation

g x h x

( ) ( )



x

n





, yielding

h x g x

'( ) ( )



h x g x

( ) '( )



nx

n1 or

(

n



)

1

x h x g x

( '( ) ( )



h x g x

( ) '( )) 1



in ,

q n

R

. The

required expression for

e x

( )

follows immediately. Parts (ii) and (iii) follow immediately from (i). □

We remark that the above proof is obtained by slightly adapting the proof for the case



1 in [ ] which on its turn was a generalization of the proof for the binary case (cf. [5]). In the next section we shall illustrate the previous theorems by a number of examples.

4. Examples

Example 1 Consider the polynomial

x

3



2

over

GF

(5)

. So,



2 and k 4, and we can take



3. We define



as a zero of the irreducible polynomial

x

2

 

x

1

over

GF

(5)

. So

3 2

1 (

1)(

1)

0



 







  



and hence



is a primitive third root of unity. Now, the three zeros of the polynomial

x

3



2

can be written as 3,

3



and

3



2, which are elements of the field

(5)( )

GF



(



GF

(5 )

2 ). We remark that Theorem 3(iv) is not applicable to the chosen root



3, since



1

 

3

GF

(5)

and

1 3(

 

n

)

. If one takes





3



, we can apply Theorem 3 (iv), since



3 is the least positive power of



which is in

GF

(5)

. Indeed, according to Theorem 3 (v) and to Theorem 4 (v), the order of

3



in

GF

(5)( )





4.3 12



.

The order of the given polynomial can also be obtained by applying Theorem 3 (ii). First we have the factorization

x

3

  

2

(

x

2)(

x

2



2

x



1)

. From [3, Table C], we conclude that ord

(

x

 

2)

4

and ord

(

x

2



2

x

 

1) 12

. Hence, ord

(

x

3

 

2)

4,12



12

. Finally, we shall verify the divisibility of

12

1

x



by

x

3



2

. We write

11

(

x



1)(

x



1)(

x

2

 

x

1)(

x

2



1)(

x

2

 

x

1)(

x

4



x

2



1)

.

Indeed, x2is a divisor of



₄

( )

x

since

x

2

  

1 (

x

2)(

x



2)

in

GF

(5)[ ]

x

, and

x

2



2

x



1

is a divisor of



12

( )

x

since

4 2 2 2

1 (

2

1)(

2

4)

x



x

 

x



x



x



x



. As for the zeros of

x

3



2

, it is clear that 3 is the zero of x2, and that

3



and

3



2

( (3 ) )





5 are the zeros of

x

2



2

x



1

. In order to determine the idempotent generators of the codes   x 2 and



x

2



2

x

 

1

we apply the rule of the previous theorem. For

g x

₁

( ) :

 

x

2

, we have h x₁( )x22x1 and hence

1 ' 2

1( ) (3.2) 1( ) 1( ) (2 2)( 2) 2 1

e x   xh x g x x x x  x  x . Similarly, we find for

2

2( ) : 2 1

g x x  x that e x₂( )x x( 22x 1) 3x2 x 2. Notice that the polynomials

e x

₁

( )

and

e x

₂

( )

are to be considered as polynomials in

R

_3,52 and not in

R

_12,51 . Using the notation introduced in the beginning of this chapter, we should denote these polynomials by



₁

( )

x

and



₂

( )

x

. The minimal constacyclic codes in this case are (x32) / (x2)x22x 1 g x₂( ) and

3 2

1

(x 2) / (x 2x 1) x 2 g x( )

       . So, in this simple case we have



1

( )

x





2

( )

x

and

2

( )

x

1

( )

x







. One can immediately verify that



1

( )

x





2

( ) 1

x



and that



i

( ) 1

x

 



i

( )

x

.

Moreover,



₁( ) ( )x



₂ x (3x2 x 2)(2x2  x 1) 0(cf. Theorem 1 (iii), (iv) and (v)). □

Example 2 Next we consider the polynomial

x

6



2

over

GF

(5)

. Again



2 and so k 4, but there is no element



in

GF

(5)

which satisfies



6





. Therefore, we define



as a zero of the irreducible polynomial

x

2



2

, and hence



2

 

2

. Applying Theorem 3(iv), with d 2 (which divides

n

),

2



 

, and h4 gives that the order of



equals 8 < 24. We also define



as a zero of the irreducible polynomial

x

2

 

x

1

over

GF

(5)

. Since



3

 

1 (





1)(



2

  



1)

0

,



is a primitive sixth root of unity. Actually, we can identify



with



3 (or with  



3), because

2

(





3)



(



  

3) 1



2

     



4



3 1 0

. So, the extension field

GF

(5)( , )

 

is isomorphic to

GF

(5)( )



. From the definition of



we have immediately that



4

 

1

and so



has indeed order 8 in

GF

(5)( )



. It also follows that



has order 24 and thus it is a generator of

(5)( )*

GF



or, equivalently, a primitive element of this extension field, contrary to the case of

3

2

x



in the previous example. According to Theorem 3 (v),

x

24



1

is a multiple of

x

6



2

and there is no polynomial

x

m



1

, m24, with the same property.

One can easily verify that the factorization of

x

6



2

into irreducible polynomials is equal to

6 2 2 2

2

(

2)(

2)(

2)

12

we find again ord

(

x

6

 

2)

8, 24



24

, applying Theorem 3 (ii). As for the factorization of

x

24



1

into cyclotomic polynomials, we can write

24 12

8 24

1 ( 1) ( ) ( )

x   x   x  x , where

x

12



1

is factorized in Example 1 into cyclotomic polynomials and next into irreducible polynomials over

GF

(5)

, while



₈

( )

x



4 2 2

1 (

2)(

2)

x

 

x



x



and



₂₄

( )

x



x

8

 

x

4

1

=

(

x

2

 

x

2)(

x

2

 

x

2)

(

x

2



2

x



2)

2

(

x



2

x



2)

are the factorizations for the ‘new’ cyclotomic ‘ polynomials.

We also determine the idempotent generators of the codes generated by the irreducible polynomials

contained in

x

6



2

. To this end we apply the relation of Theorem 5 . We find the following results: (i) g x₁( )x22; h x₁( )(x2 x 2)(x2 x 2)x42x2 4; 3 2 4 2 1( )x 3 (4x x 4 )(x x 2) 2x x 1



      . (ii) g x₂( )x2 x 2; h x₂( )(x22)(x2 x 2)x4x34x2 2x4; 3 2 2 5 4 2 2( )x 3 (4x x 3x 3x 2)(x x 2) 2x x 2x 2x 1



           . (iii) g x₃( )x2 x 2; h x₃( )(x22)(x2 x 2)x44x34x23x4; 3 2 2 5 4 2 3( )x 3 (4x x 2x 3x 3)(x x 2) 3x x 2x 3x 1



           .

The primitive idempotent polynomials are:



₁( )x 3x4x22,



₂( )x 3x5x42x22x2 and



₃( )x 2x5x42x22x2. As one can see

3 1 ( ) 1 i s x



 



(cf. Theorem 1 (iv)). In a completely similar way we determine the idempotent generators for the case n6,

q



5

,

3



 , and hence k 4. Firstly, we obtain the factorization of

x

6



3

into irreducible polynomials over

GF

(5)

:

x

6

 

3

(

x

2



2)(

x

2



2

x



2)(

x

2



2

x



2)

. Next, by applying Theorem 1, we derive the following idempotent polynomials:

(i)



₁( )x 3x4x21 for g x₁( )x22; (ii)



₂( )x   x5 x42x2 x 1 for g x₂( )x22x2; (iii)



₃( )x x52x4x2 x 2 for g x₃( )x22x2.

We also study the case n6,

q



5

,



 1, and hence k 2, with the factorization

6 2 2

1 (

2)(

2)(

2

1)(

2

1)

x

  

x

x



x



x



x



x



. The results are:

(i)



₁( )x  2x5x42x3x22x for

g x

₁

( )

 

x

2

;

(ii)



₂( )x 2x5x42x3x22x for

g x

₂

( )

 

x

2

; (iii)



₃( )x  2x5x4x3x22x1 for g x₃( )x22x1;

13

Example 3 Next we take n8,

q



3

,



 1, and hence k2. The factorization of

x

8



1

into irreducible polynomials over

GF

(3)

is

x

8

 

1 (

x

4



x

2



1)(

x

4



x

2



1)

. If



is defined as a zero of the first polynomial, i.e. if



4





2

 

1 0

, then



8

 

1

and



16



1

. We observe that the two irreducible polynomials are not primitive. Actually, both have order 16 and so, by Theorem 3, the order of

x

8



1

equals 16 as well. This is in agreement with Theorem 3 (iv), since hd 8.2 16 is divisible by 8. To obtain a primitive

8

th root of unity, we extend

GF

(3)

by a zero



of the polynomial

x

4

 

x

1

which is irreducible in

GF

(3)[ ]

x

, and which has order (or exponent) 80 (cf.[3, Table C] ). So, if we extend

GF

(3)

by



, then



is a primitive element of

GF

(3)( )



4

(



GF

(3 ))

. We can identify

5

:

 



, since



20





10

 

1 0

, and so



is a zero of

x

4



x

2



1

. Similarly, if we define

25

' :







, then



' is a zero of

x

4

 

x

2

1

. Furthermore, if

 

:



2

(





10

)

, then



is a primitive

8

th root of unity and the eight solutions of the equation

x

8

 

1 0

can be written as



i, 0 i 7. By applying Theorem 1, we find for the idempotent generator of the negacyclic code



g x

₁

( )



:=

4 2

1

x

x





 

the polynomial e x₁( )  x6 x21. Similarly, we find for the negacyclic code

2

( )

g x





:=



x

4

  

x

2

1

the idempotent generating polynomial e x₂( )x6x21. □

5. A relation between idempotent generators of constacyclic and cyclic codes

In this section we derive a simple relationship between the idempotent generating polynomial over

( )

GF q

of a constacyclic code of length

n

and those of certain cyclic codes of length N , with

( , )

n q



( , ) 1

N q



.

Theorem 6 Let n be an integer



1

, q a prime power with

( , ) 1

n q



and let





GF q

( )*

have order k. Let

n

x





be a divisor of

x

N



1

in

R

_Nq_,1, with

( , ) 1

N q



. Let furthermore

g x

( )

be an irreducible polynomial of degree m, dividing

x

n





. If

e x

_

( )

is the idempotent generator of the



-constacyclic code



g x

( )



_ in

R

_nq_, and

e x

( )

the idempotent generator of the cyclic code



g x

( )



in

R

_Nq_,1, then

( )

( )

e x

_



e x

mod

x

n





. In particular this is true for N:e, where e is the order of the polynomial

x

n





over

GF q

( )

.

14

(

x

n





)

xh x g x

'( ) ( )



nx h x g x

n

( ) ( )



(xN 1)xh x g x_'( ) ( )Nx h x g xN ( ) ( ). By applying Theorem 5, we can write for this equality

(

x

n





)(

Ne x

( )



l x

(

N



1))



nx x

n

(

N



1)

=(xN 1)(n e x



_( )l x'( n



))NxN(xn



), for some integers l and l'. Next, we divide both sides of this equality by

x

n





, giving ( ) ( N 1) n ( ) ( ) ( ) '( N 1)

Ne x l x  nx t x n t x e x



_ l x 



Nx

N.

We consider this relation as an equality modulo

x

n





. This implies that

x

n





and

x

N



1

. We also know that ekn by Theorem 4 (v) and that Nbe for some positive integer b by [3, Lemma 3.6]. So we can write Nan, with k∣

a

. Hence, modulo

x

n





, we have that

( )

(

N

1) / (

n

)

(

an

1) / (

n

)

t x



x



x







x



x





= 1

(

/ )

1

/

1

n a n

x

x













1 0

1

(

)

/

n a i i

x

a







 







.

Substituting these results yields modulo

x

n





the relation

e x

_

( )



e x

( )

. In particular, we can take :

N e, since

( , ) 1

e q



. This is because k∣

q



1

, hence

( ,

k q

 

1) 1

and

( , )

e q



(

kn q

, ) 1



, and thus

e

satisfies the condition of the theorem. □

Example 4 We consider the codes of Example 3, so n8,

q



3

,



 1, k2. Take

g x

( ) :



x

4



x

2



1

.

Then in

R

16,31 we have

4 2 8 12 10 8 4 2

( )

(

1)(

1)

1

h x



x



x



x

 

x



x

 

x

x



x



. Applying Theorem 1 gives

e x

( )

 

x

14



x

10

 

x

8

x

6



x

2. By taking this expression mod

x

8



1

and applying

Theorem 6, we obtain e_1( )x   x6 x21.

Next we consider the codes of Example 2, so n6,

q



5

,



2, k 4. Take

g x

( ) :



x

2



2

. Then in

R

_24,51 we have

h x

( ) :



x

22



2

x

20



x

18



2

x

16



x

14



2

x

12



x

10



2

x

8



x

6



2

x

4



x

2



2

.

Again, applying Theorem 1 gives for the idempotent generator in

R

1_24,3 the expression

22 20 18 16 14 12 10 8 6 4 2

( )

2

2

2

2

2

2

e x



x



x



x



x



x



x



x



x



x



x



x



, which modulo

x

2



2

provides us with e x₂( )2x4x21 as the idempotent generating polynomial in

R

_6,52 . (cf. also Example 2). □

6. More examples

We present a few more examples as illustrations of the previous theorems and for future use. Example 5 Consider the polynomial

x

7



3

over

GF

(5)

. So, n7,

q



5

,



3, and hence k4. One can

15

7 6 5 4 3 2

3

(

2)(

2

4

3

2

4)

(

2) ( )

x

  

x

x



x



x



x



x



x

  

x

f x

. The order of

2



GF

(5)

is 4, so

1

4

e



and x2 is a divisor of

x

4



1

. If



is some zero of

f x

( )

, then



7



3

. It is well known that if



is a zero of an irreducible polynomial

P x

( )

over

GF q

( )

, the elements

 

q

,

q2

,....,



qm1 are the other zeros of

P x

( )

, where

m

is the degree of

P x

( )

. In this case the successive

q

-powers are



,



5,



52



2



4,



53



3



6,



54



3



2,



55



4



3,



56





. We conclude that m6 and that the second factor

f x

( )

in the above decomposition is irreducible over

GF

(5).

It also follows that F:

GF

(5)( )





GF

(5 )

6 is the splitting field of

x

7



3

and consequently,

x

7



3

divides

6

5 1

1

x





. The order of

x

7



3

is equal to the least common multiple of the orders of

f x

( )

and x2 (cf. Theorem 3 (ii)) and hence it is a divisor of

5

6



1

. The order of

f x

( )

is equal to the order of



in the group

GF

(5)( )*



. We remark that



is not a primitive element of this field, since

6

( 1)



f

(0)



4

is not a primitive element of

GF

(5)

(cf.[ N.R., Theorem 3.18]). It also follows from Theorems 3 and 4 that

GF

(5)( )



contains a primitive

7

th root



of unity and that we can identify the field

F

in these theorems with the extension field

GF

(5)( )



. Since



7 is the lowest power of



which lies in

GF

(5),

Theorem 3 (iv) yields that the order of



is 7.4=28. Indeed, starting from the defining polynomial

f x

( )

of



, we obtain by a simple calculation that



7



3

, and hence

14

4





and



28



1

. So, the order of

x

7



3

is equal to

e



e e

1

,

2



4, 28



28

. Of course, this

value also follows from Theorem 4 (v). To illustrate Theorem 4 (ii) and (iii), we remark that



₁



2

is a zero of the first irreducible polynomial and



₂





of the second. So, according to Theorem 4 (ii)

2







is the generating element of the subgroup of

GF

(5)( )



of order

e

( 28)



. Indeed, we have

28 28 28 28 4 4

(2 )

2

3 .3

1















. Furthermore,

 



28/7





4





4 is a primitive

7

th root of unity in

F

. The subgroup

H

:





4



{1,

   

4

,3 ,3

5

, 4

2

, 4

 

6

, 2

3

}

contains all zeros lying in

F

of the polynomial

x

7



1

. More generally, the coset H_i 



iH contains all zeros in

F

of the polynomial

x

7



2

i, for 0 i 3. More precisely, the cosets

H H

₁

,

₂ and

H

₃ contain the zeros of

7

3

x



,

x

7



4

and

x

7



2

, respectively. □

Example 6 Take n5,

q



7

and



2. So, k 3. We choose



4 which is a zero of

x

5



2

. There is a

j

,

0

 

j

5

, with



j



GF

(7)

, i.e.

j



1

, so the second part of Theorem 3 (iv) gives ord (



) = 3



3.5 = 15. Furthermore,

GF

(7)( )





GF

(7)

does not contain a primitive 5th root of unity, since 5 is not a divisor of ∣

GF

(7)*

∣ = 6. The factorization of

x

5



2

into irreducible polynomials in

(7)[ ]

GF

x

is

x

5

  

2

(

x

4)(

x

4



4

x

3



2

x

2

 

x

4)

.

16

x

4



4

x

3



2

x

2

   

x

4

(

x



)(

x





7

)(

x





72

)(

x





73

)

=

(

x





)(

x



2



2

)(

x





4

)(

x



4



3

)

, where



now stands for some arbitrary zero of that polynomial. Extending

GF

(7)

by this



yields a field

F

:



GF

(7)( )



of size

7

4, since



is defined by an irreducible polynomial over

GF

(7)

of degree 4. The condition of Theorem 3 (iv) is satisfied now and hence, the order of



is 3.5 = 15. From Theorem 3 (v) it follows that ord (

x

5



2

) = 15 and also that we can take

 



3. So, the zeros of

x

5



2

are



,



4,



7

( 2





2

)

,



10

( 4)



and



13

( 4





3

)

.

Next, we take n3with the same values for

q

and



, and so k3 again. In this case

GF

(7)

contains a primitive

n

th

( 3 )



rd root of unity, i.e.





2

. Like before, Theorem 4 (v) yields

immediately that ord

( )





3.3 9



. Unlike in the former case n5, the polynomial

x

3



2

has no zeros in

GF

(7)

. We factorize this polynomial into linear polynomials over its splitting field

(7)( )

GF



as follows

x

3

  

2

(

x



)(

x





7

)(

x





49

)

 

(

x



)(

x



4 )(



x



2 )



.

The size of this extension field is equal to

7

3. Furthermore, since

 

,

2



GF

(7)

, we can apply Theorem 3 (iv). One can easily verify that

x

3



2

is a divisor of

x

9



1

by writing

x

9

  

1 (

x

1)(

x





3

)(

x





6

){(

x





)(

x





7

)(

x





4

)}{(

x





2

)(

x





5

)(

x





8

)}

=

(

x



1)(

x



2)(

x



4)(

x

3



2)(

x

3



4)

.

The fact that 9 is the smallest value of

n

such that

x

3



2

divides

x

n



1

, can be understood by realizing that the only possible candidate less than 9 is n6. Remember that

x

a



1

is a divisor of

1

b

x



if and only if

a

divides b. But

x

6

 

1 (

x

3



1)(

x

3



1)

ruling out

x

3



2

as a divisor. □

7. Cyclonomic polynomials for constacyclic codes

In order to deal with idempotent generating polynomials for constacyclic codes in a similar way as for cyclic codes, we introduce the notion of cyclonomic polynomial for



-constacyclic codes. We shall do this in such a way that this notion gets its usual meaning for



1, i.e. in case of cyclic codes. First, we present the well-known notion of cyclotomic coset mod

n

. Let

s

be some integer with

0  s n 1. Let

m

_s be the smallest positive integer such that

_sq

ms



_s

_mod

_n

_{, then the cyclotomic}

coset modulo

n

to which

s

belongs is defined as n: { , ,...., ms 1}

s