Orthogonality and Conjugation for the Idempotents of Constacyclic Codes

Tilburg University

Orthogonality and Conjugation for the Idempotents of Constacyclic Codes

van Zanten, A.J.

Publication date:

2016

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Citation for published version (APA):

van Zanten, A. J. (2016). Orthogonality and Conjugation for the Idempotents of Constacyclic Codes. Tilburg University.

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1

Orthogonality and Conjugation

for the Idempotents

2

Abstract

3

Contents

1. Introduction p. 4

2. Orthogonality relations for idempotents p. 4

3. Conjugated constacyclotomic cosets p. 18

4. Conjugated irreducible polynomials p. 22

5. Conjugated constacyclonomials p. 26

6. Constacyclotomic cosets p. 33

7. Constacyclotomic (di)graphs p. 36

4

1. Introduction

This report is a continuation of the study of the generating and irreducible polynomials of

constacyclonomials of constacyclic codes as described in [1, 2, 3]. Codes generated by polynomials which divide

x

n





,





GF q

( )*

, are called (



-) constacyclic codes. This family of codes generalizes the well-known class of cyclic codes when



1. Let n ( )( )

s s S

x



P x



 



be the decomposition of

x

n





into monic irreducible polynomials over

GF q

( )

,

( , ) 1

n q



. The code  f x_s( ) with

f x

_s

( ) : (



x

n





) /

P

_s( )

( )

x

is called a minimal or irreducible constacyclic code, contrary to the code P_s( ) ( )x  which is called a maximal constacylic code. The idempotent generating polynomials of the minimal constacyclic codes are denoted by



_s( )x and those of the maximal constacyclic codes by



_s( )x . The polynomials



_s( )x are occasionally called primitive idempotent polynomials. For more introductory remarks and references we refer to the introductions in [1, 2, 3].

2. Orthogonality relations for idempotents

In order to keep our notation as simple as possible we shall drop the fixed parameters n and q from all variable names (like we did on earlier occasions), and write

T



:



T

n q, ,,

S



:



S

n q, ,,

, , : n q t t C C , : n q, t t C C , ( ) : n q, , ( ) t t e x e  x , ( ) : n q, , ( ) s s c x c  x , etc. From [1,2,3] we have the following factorization of

x

n





into monic irreducible polynomials over

5

: q( )

k ord



, satisfying



n





. If



i is a zero of the irreducible polynomial Pt( )( )x 

, then

(



i q

)





j for some

j

is a zero of the same polynomial . It follows that we can take for

T

 a subset of

{0.1,....,

n



1}

, such that the index t of each polynomial Pt( )( )x



corresponds to one of its zeros



t. Usually, we take the minimal value possible. More precisely, from [2, Theorem 6] it follows that if



t is a zero of Pt ( )x



, then all zeros of that polynomial can be written as



c, where c runs through the constacyclotomic coset C_t,

t



{0,1,....,

n



1}

, (cf. [3, Definition 2]). We shall write the polynomial P_t( ) ( )x explicitly as

( )

( )

t _,1 t 1

....

_, t m m t t t m

P



x

x



p x

 

p

  





 

. (2)

From the lines above it follows that the degree m_t of Pt ( )x 

is equal to the size of C_t. As for the set

T

, we proved (cf. [3, Corollary 19]) that its size is equal to the size of

S



: {



s

∣c_s( )x is an element of a maximal independent set of constacyclonomials}. For reasons of convenience we therefore introduce the variable

n

₀

:



∣

T

∣= ∣

S

∣. (3)

Since

 



k,

k

:



ordq( )



, an alternative expression for the n zeros of n

x





is



ik1,

0

  

i

n

1

(cf. [1, Theorem 3 (v)]). In the next we shall occasionally need the zeros of

x

n





j,

0

  

j

k

1

. Therefore, we sometimes shall use the alternative representation of the zeros, since it makes dealing with them easier (cf. the remarks right after Theorem 13 and Example 10 in [2]). Actually, all zeros of the polynomial

x

kn



1

are covered in this way at the same time. By doing so, the set

T

 will be a subset of

{0,1,....,

kn



1}

.

In [2,3] the notion of constacyclonomial was introduced. For our convenience we repeat the most relevant facts. For the details we refer the reader to [2,3]. The polynomial

csn q, , ( )x xs xsq .... xsqms 1        mod

x

n





is called a constacyclonomial if n q, , ( )q n q, , ( ) s s

c  x c  x and if it is not the zeropolynomial. This polynomial can be written as a polynomial in

GF q x

( )[ ]

n q, , _{( )} d0 c0 d1 c1 _.... dms1 cms1

s

c  x 



x 



x  



 x  ,

with c_iC_sn q, , 0 i m_s, and

d

₀



0

. A necessary and sufficient condition for c_sn q, ,( )x to be a constacyclonomial is

x

sqms

x

s





, or equivalently

_{s q}

(

ms

1)

0



 

mod

kn

, where

k



ord_q( )



.

Since



q





for all elements





GF q

( )

, the polynomial



c

_sn q, ,

( )

x

also satisfies the definition requirement for constacyclonomials, i.e. (



c_sn q, ,( ))x q 



c_sn q, ,( )x . Therefore, from now on we shall call all polynomials



c_sn q, ,( )x satisfying

x

sqms

x

sq





mod

x

n





constacyclonomials. The special case





1

could be dubbed as a monic constacyclonomial , similarly to monic (irreducible)

polynomials. We shall often identify the linearly dependent constacyclonomials

c

sn q, ,

( )

x

 and , , ( ) n q s c  x

6

We remind the reader that we defined

S

 such that if c_j is not self adjoint and if

j



S

, then also

n

 

j

S

. For our convenience we therefore introduce the notion of adjoint constacyclonomial. If

c

_j

( )

x

is not self adjoint, we define the adjoint constacyclonomial

c

_j*

( )

x

as

c

_j*

( ) :

x



c

_{n j}_

( )

x

, (4) while in case that

c

_j

( )

x

is self adjoint we just take

c

_j*

( ) :

x



c x

_j

( )

. So, according to this definition, in all cases the adjoint constacyclonomial is again a (monic) constacyclonomial. In [2] we we

occasionally wrote

c

__j

( )

x

instead of

c

_{n j}_

( )

x

. This notation will not give rise to confusion as long as we stick to the ring Z_n. However, in expressions like



s, one should keep in mind that it stands for

n s





, and hence

 

s s 



n(



o 1). Similarly, we define the adjoint irreducible polynomialPt( )*( ) :x P(t 1)( )x

 



 . So, the zeros of

( )*

( ) t

P x are zeros of

x

n





1 and P_t( )* ( )x itself is a factor of this binomial. It is well known that the inverses of the zeros of Pt( )( )x



are the zeros of its monic reciprocal. ButPt( )*( )x 

is not always a factor of the monic reciprocal of

x

n





1 which is

x

n





. Hence, we can say that the adjoint of

( )

( ) t

P x is equal to its monic reciprocal if and only if

 



1, i.e. if



 1. From [2, Theorem 13] it now follows that the

n

₀ constacyclonomials

c

j



,

j



S

, constitute an orthogonal basis of polynomials in the algebra A with respect to the inner product

1 0 ( , ) ( ) ( ) n i i i p q _ p



q



  



, i.e. : { s s( ) s S A c x  



 



∣cs C , s GF q( )} _ 

_

_ , (5) ( *, ) aj _, j k j j k c c  nm

 

, (6) where aj: 1 if

c

j

( )

x



is not self adjoint and : ( [mj/2] 1) / j

a  j q  n if it is self adjoint. Occasionally, we shall write

c

j

( )

x



 for the adjoint constacyclonomial

c

j*

( )

x



instead of

c

n j

( )

x



 ,

where



j

stands for the integer in

{0,1,....,

n



1}

which is equal modulo n to



j

. We emphasize that without that condition

c

__j

( )

x

is not a constacyclonomial and not even an element of

R

_{n q}_, .

Example 1

We compute for each value of



an orthogonal basis for the vector space

C

6,5, as described above.

7

4





4 0( ) 1 c x  , c x₁4( )x1x5, c x₂4( )x2x4, c x₃4( )x3.

All these constacyclonomials are self adjoint and hence, for all



and

j

we have

c

_j*



c

_j. As a couple of illustrations of eq. (6) we compute:

5 5 2* 2 1 5 1 5 2 6 10 1 1 2 0 0 ( , ) ( )( ) _x i ( 2 ) _x i i i c c x x x x _{ } x x x _{ }   



  



  =

6.4



24



4

, 5 5 2* 2 2 4 2 4 4 6 8 2 2 2 0 0 ( , ) ( 2 )( 2 ) _x i ( 4 4 ) _x i i i c c x x x x _{ } x x x _{ }   



  



  =

6.4.2



48 3



, 5 5 4* 4 3 3 6 3 3 4 0 0 ( , ) _x i _x i 6.4 24 4 i i c c x x _{ } x _{ }   







   .

Applying (6) yields the following outcomes for the above inner products:

2* 2 1

1 1 2

(c ,c ) 6.2.2 244, (c₂2*,c₂2)₂ 6.2.22 483 and (c₃4*,c₃4)6.1.41244, which agree with the direct calculations.

We also compute the inner products of c₅2*( )x x5x1 and c x₄2( ) with themselves and obtain

2* 2

5 5 2

(c ,c ) 4 and (c₄2*,c₄2)₂ 2. Indeed, eq. (6) provides us with (c₅2*,c₅2)₂ 6.2.25 4 and

2* 2 4

4 4 2

(c ,c ) 6.2.2 2. The last result can also be obtained by realizing that 3c₂2 c₄2, and so

2* 2 2 2* 2

4 4 2 2 2

(c ,c ) 3 (c ,c )9.32. □

It follows that each element p

A

, i.e. the vector space consisting of all polynomials in

R

n q, 

which satisfy p x( )q  p x( ), (cf. [2, Theorem 13 (i)]) can be developed w.r.t. the basis {c_s∣

s



S



}

, so ( ) s s( ) s S p x c x  



 



, ( *, ) / as s cs p nms 







,

s



S

, (7)

where we used

m

_s



m

_{n s}



m

_s in the case that cs( )x 

is not self adjoint. In particular, we can consider p x( )



_t( )x ,

t



T

, a so-called primitive idempotent polynomial which generates the constacyclic code ( ) /{ } ( ) u u T t P x   





 of length n over

GF q

( )

, which we call a minimal or

irreducible code, thus generalizing that notion as known in the theory of cyclic codes. This idempotent

satisfies

 

_t( i)1 if



i (or



ki1) is a zero of P_t( ) ( )x , while

 

_t( i)0 if



i is not a zero of Pt( )( )x



, for

0 i

 

n

. Another special case is the idempotent t ( )x 1 t ( )x

 



 



which generates the maximal code P_t( ) ( )x .

Theorem 1

Let

C

be the constacyclic code of length n generated by some polynomial in

R

_nq_, and let

, ,

( )

n q

e x





C

 be its idempotent generator. Let furthermore {cs( )x 

8

of constacyclonomials in ,

q n

R

_ for the chosen values of n and q.

(i) There exist coefficients



_s



GF q

( )

such that ( ) _{s s} ( ) s S

e x c x





_









and they are uniquely determined by the relation ( *, ) / as

s cs e nms

 









.

(ii) The coefficients



_st in the expression for the primitive idempotent generator _t ( ) _st _s( ) s S x c x   





 



,

t



T

, are equal to _'s

/

_'s s t t s a t t

m

p

m

n

  





 

, where p_t_'s is the one but highest coefficient of xin the irreducible polynomial P_t_'s( )x which is a factor of

x

n





s while m_tn_',s is its degree, and where

'

( , )

t



t k s

if c_s is self adjoint, whereas

t

'



t k

( ,

 

s

)(

t k n s

( ,



))

if c_s is not self adjoint.

(iii) The coefficients



_st in (ii) can also be written as s / s s t t s a st st m p m n   





  , where the index

st

has

to be taken with respect to the zero



s of

x

n





s.

(iv) The inproduct of a pair of primitive idempotents



_t and



_u is equal to

(

t

,

u

)

m

t t u,

   

 





. (v) If t ( ) st s( ) s S x c x   





 



and u ( ) su s( ) s S x c x   











are two primitive idempotent polynomials, then _{s s}t u_{s t t u,} s S n w m  

 

_



 



, with as s s w m



.

(vi) If s and r are both elements of

S

, then _s 1 _st _rt _{s, r}

t T t nw m  

 



  



. Proof

(i) This follows immediately from eq. (7). (ii) We know that

 

_t( i) is equal to 1 if



i is a zero of the irreducible polynomial P_t( )x , while

it is equal to 0 otherwise. Moreover, c_s(



i)(



i s) (



i sq) .... is constant for

i

-values from the same constacyclotomic coset, since if

i

and i' are both from Ctn q, ,



, then there exists an exponent

j

such that



i'



(



i q

)

j . Hence, 1 * * * 0 ( , ) ( ) ( ) ( ) n i i t s t s t t s i c



 _ c

  

 m c 



  



 . Firstly, we

assume that c_s( )x is self adjoint, and so c_s*( )x c_s( )x . Applying (7) and [2, Theorem 13 (v)], we can write (c_s*,



_t)_= m m pt s tn_', s /mtn_', s     , with

t

' :



t k s

( , )

. So, _', s / _', s s t t n n s a t t m p m n   





  , with [ /2] ( ms 1) / 2 s

a s q  . Next, we assume that c_s( )x is not self adjoint, and so c_s*( )x c_{n s}_ ( )x . In this case we find by applying [2, Theorem 13 (v)] that (c_s*,



_t)_  m m_t _{n s} p_tn_',s /m_tn_',s, with

'

( ,

)

t



t k n s



. Since m_{n s} m_s, it follows that _', s / _', s n s t t n n s a t t m p m n   





   , with a_{n s} 1, since n s

c_ is not self adjoint, just like c_s. (iii) The s-power of any zero of P_t( ) ( )x is a zero of the irreducible polynomial P_st(s)( )x which is a

divisor of

x

n





s. The result now follows immediately from (ii). (iv) We have 1 , 0 ( , ) ( ) ( ) n i i t u t u t t u i m      

 



   





9

(v) Substituting the given expressions in the inproduct yields _{' '}

, ' ( _t , _u ) _st _su( _s, _s) s s c c      

 





 

= ( , ) as t u t u s s s s s s s s s c c _ n m

 

_{ } 

  

_





, and the relation follows by applying (iv). (vi) By using (4) and (6), we can write

1 , 0

(

)

(

)

(

)

(

)

s n a t t i i t s r s r s s r i t T

m c

c

c

c

nm

  

_



_

 

_



_

_{ }

   









. Substitution in the lhs of this expression ( t) as t

t s s s m c _



nm

 

and ( t) ar t t r r r m c 



nm

 

_ yields _, r a t t r s r s r t T t

nm

m

 

   

_ 





, and so the relation (v) follows. □

Example 2

For

n



6

and

q



5

we have the following factorizations into irreducible polynomials and next into factors

x





i where



is a zero of

x

2



2

x



1

which is of order 24:

6 2 2

1 (

1)(

1)(

1)(

1)

x

  

x

x

 

x

x

 

x

x



=

(

x





0

)(

x





4

)(

x





20

)(

x





8

)(

x





16

)(

x





12

)

; 6 2 2 2 1 5 9 21 13 17

2

(

2)(

2)(

2)

(

)(

)(

)(

)(

)(

)

x

 

x

 

x

x



x

   

x

x



x





x





x





x





x





; 6 2 2 2 3 15 7 11 19 23

3 (

2)(

2

2)(

2

2)

(

)(

)(

)(

)(

)(

);

x

 

x



x



x



x



x

  

x



x





x





x





x





x





6 2 2 2 10 6 14 22 18

4

(

2

1)(

2)(

2

1)(

2)

(

)(

)(

)(

)(

)(

).

x

 

x



x



x



x



x



x

  

x



x





x





x





x





x





In [2, Theorem 5 (iii)] we stated that for any

j

,

0

  

j

k

1

, the n zeros of

x

n





j can be written as



j ik ,

0

  

i

n

1

, where



is a zero of order

kn

. By defining

 

:



k, one can equally well write the n zeros as

 

j i,

0

  

i

n

1

. The following example provides us with the primitive idempotents of the cyclic codes generated by the irreducible factors of

x

6



1

.

1





,

k



1

,

kn



6

Let



be a primitive

6

th root of unity, e.g. a zero of

x

2

 

x

2

. Then it follows from the relations at

the bottom of the previous page (with

 



4) that x6 1 P₀(1)( )x P₁(1)( )x P₂(1)( )x P₃(1)( )x , with

(1) 0 0 ( ) 1 P x    x x



, P₁(1)( )x x2  x 1 (x



1)(x



5), (1) 2 2 4 2 ( ) 1 ( )( ) P x x   x x



x



, P₃(1)( )x    x 1 x



3.

We define

S

1



{0,1, 2,3}

and

T

1



{0,1, 2,3}

. All (consta)cyclonomials are self adjoint, and since

1

k , we have d_s (kn s, ) / ( , ) 1n s  for all

s



S

6,5,1. In this case the expression for the coefficients



_st in



t1( )x simplifies to

',1 ',1 n t t t s n t

m p

n m



 

, which reduces to 1 st t st

p

m

m



in

GF

(5)

. Applying this to



₀1( )x gives

1 0 0 0 0 0 0 1 2 3 0 0

p

m

m















 



 

1. 1

= 1, and hence, using the expression for (consta)cyclonomials in Theorem 1 (ii) , we obtain

1 1 2 3 4 5

0( )x 1 x x x x x

10 Similarly, we find for



₁1( )x the coefficients

1 1 0 0 1 0

2. 1

2

p

m

m



 

   

, 1 1 1 1 1 1

1

2

1

2

p

m

m



 

 





, 1 1 2 2 1 2

p

m

m



 

=

2

1

2



= 1, 1 1 3 3 1 3

p

m

m



 

=

2

1

2

1



 

. So,



₁1( )x   2 x1 x22x3x4x5.

In the same way, omitting the explicit calculations, we obtain for



₂1( )x its coefficients



₀2 2,

2

1 1



  ,



₂2  1,



₃2 2, and so



₂1( )x   2 x1 x22x3x4x5, while for



₃1( )x we get

3

0 1



 ,



₁3 2,



₂3 2,



₃3  1, and so



₃1( )x   1 x1 x2x3x4x5. As for these and similar calculations one should keep in mind that products like st have to be carried in the ring

Z

₆. The above expressions for



t1( )x satisfy the idempotent relation

1 2 1

( ) ( )

t x t x







, and turn out to coincide with the expressions obtained by applying the general formula for idempotent generating polynomials

e x

( )



(

n



)

1

xh x g x

'( ) ( )

(cf. [1,2]). E.g. by taking h x( )P₁(1)( )x x2 x 1 and

6 2 4 3

( )

(

1) /

1

1

g x



x



x

  

x

x

  

x

x

, we get again

xh x g x

'( ) ( )



x

5

 

x

4

2

x

3

  

x

2

x

2

. Instead of writing the zeros as



i,

0

 

i

5

, one can also represent them by



i,

0

 

i

5

, where



is some zero of

x

6



1

and



some primitive

6

th root of unity. E.g. one could take

 



, in which case the indexation of the irreducible polynomials is: P₅(1)( )x  x 1, P₀(1)( )x x2 x 1,

(1)

1 ( )

P x 

x

2

 

x

1

, P₂(1)( )x  x 1. Actually, this last way of representing the zeros is a special case of the convention of taking

 



k, with



a zero of order

kn

. □

Next, we shall try to do the same for



 

1

and the same values for n and q.

Example 3

Take



 

1

, hence

k



2

and

kn



12

. Let

 

(



2

)

be a zero of order 12, and choose

 



2. Then we have the following factorization

into irreducible polynomials x6 1 P₀( 1) ( )x P₁( 1) ( )x P₃( 1) ( )x P₄( 1) ( )x , with

( 1) 2 1 5 0 ( ) 2 1 ( )( ) P x x  x  x



x



 

(

x



0

)(

x





2

)

, ( 1 3 1 1 ( ) 2 P x    x x



 x



, ( 1) 2 3 5 3 ( ) 2 1 ( )( ) P x x  x  x



x



, ( 1) 4 4 ( ) 2 P x    x x



.

Here, we defined

S

1



{0,1, 2,3}

and

T

1



{0,1,3, 4}

. However, when applying the expression for s



in Theorem 1 (ii), also polynomials Pst(1)( )x will play a role having as zeros the s-powers of the

zeros of P_t( 1) ( )x . Therefore, the index st does not have its usual meaning with respect to



, but to s

11

a subset of

[0,

kn



1]

. To this end, one has to reformulate a couple of properties of the constacyclonomials cs( )x



in terms of



-powers. □

Theorem 2

Let



be a zero of

x

n





of order kn.

(i) For the inner product

( , )

p q

_,

p q

,



C

n q, ,, one can write

1 1 1 0 ( , ) ( ) ( ) n ik ik i p q _ p



q



    



. More

generally, for

p q

,



C

n q, ,j, one can write

1 0 ( , ) s ( ) ( ) n s ik s ik i p q _ p



q



    



.

(ii) The cyclonomials

c c

j

,

k

C

n q, ,

 

_



satisfy the orthogonality relation ( *, ) aj _,

j k j j k

c c _ nm

 

.

(iii) For any

s



0

and for any j, one has

1 0 ( ) 0 n j ik s i c



   



, while 1 0 0 ( ) n j ik i c



n    



.

(iv) For any

s



S

, for any

i



[0,

n



1]

and for any

j



[0,

k



1]

, one has

(

)

.

js js j ik is s s is

p

c

m

m

  





_{ }

(v) Let

e x



( )

be some polynomial in

C

n q, ,, then one can write

( )

_{s s}

( )

s S

e x

c

x

 

_

 





with

( )

s

GF q





and ( *, ) / as s cs e nms  







.

(vi) If



_t( )x ,

t



T

, is the primitive idempotent generator of the constacyclic code

( ) \ ( ) i i T t P x    



, then _t ( ) _st _s( ) s S x c x   





 



with js js s t t st s a st m p n m   





  . Proof

All relations follow from [3, Theorem 5 (v), (ix), (x), (xi)] and from Theorem 1. □

Example 4

Again

n



6

,

q



5

,



 

1

and so

k



2

,

kn



12

. We now shall determine the idempotents

1

( )

t x





,

t



T

1, by applying Theorem 2 (iv). Again we take for a primitive

12

th root of unity



a zero of

x

2



2

x



1

, which gives rise to the following factorization of the irreducible polynomials:

12

We now apply for the above parameter values the expression

1 ( 1) ( 1)

( 1)

s s s s s t t st t st s a a st st

m

p

m

p

n

_m

_m

  





  

 

 



,

where we omitted the *-symbol, since all irreducible polynomials are self adjoint.

13

2

s



,

m

₂

 

2

a

₂



2

, 29

11

1

11



 

 

,

3

s



,

m

₃

 

1

a

₃



1

, ₃9

1

2

2

1 1



 



 



, 1 1 5 2 4 3 9 ( )x 1 2(x x ) (x x ) 2x



 _{     } .

In order to check the correctness of these results, we also compute the primitive idempotents by the general equations (cf. [1, Theorem 5])



_t1( )x  1



_t1( )x ,



_t1( )x (n



)1xh x g x'( ) ( ), g x( )P_t( 1) ( )x . We find the following polynomials:

( 1) 2 1 ( ) 2 1 P x x  x ,

h x

( )



x

4



2

x

3



2

x



1

,

h x

'( )



4

x

3

 

x

2

2

, 1 1 ( )x



 _ 2 3 4 5

2 2



x



x

  

x

x

2

x

, ( 1) 3 ( ) 2 P x  x ,

h x

( )



x

5



2

x

4

 

x

3

2

x

2

 

x

2

,

h x

'( )

 

2

x

3



2

x

2

 

x

1

, 1 2 3 4 5 3 ( )x 1 2x x 2x x 2x



 _{     } , ( 1) 2 7 ( ) 2 1 P x x  x ,

h x

( )



x

4



2

x

3



2

x



1

,

h x

'( )

   

x

3

x

2

2

, 1 2 3 4 5 7 ( )x 2 2x x x x 2x



 _{     } , ( 1) 9 ( ) 2 P x  x ,

h x

( )



x

5



2

x

4

 

x

3

2

x

2

 

x

2

,

h x

'( )



2

x

3



2

x

2

 

x

1

, 1 2 3 4 5 9 ( )x 1 2x x 2x x 2x



 _{     } .

So, there is fully agreement between the two methods in this case.

The matrix which has as columns the representing vectors with respect to the basis of negacyclonomials c_s1( )x ,

s



{0,1, 2,3}

t 1 3 7 9 m_t1  2 1 2 1 s

m

_s ws 

2

1

2

1

2

2

2

2

1

1

1

1

1

2

1

2







_

_

















_

_







0 1 2 3 1 2 2 1 1 3 2 1  In order to check Theorem 1 (iv) and (v), we determine the weight factors : as

14 7 9 s s s s w

 





1.2.1 ( 2).2.2 2.1.( 1) ( 1).( 1).( 2)

 



  



 

0

, 9 9 s s s s w

 





2 2 2 2

1.1

 

( 2)2



2.( 1)



  

( 1)( 2)



1

.

Both results satisfy the relation of Theorem 1 (iv). Notice that m₉11. We also present a few examples to illustrate Theorem 1 (v):

0 1 1 1 3.2( 2) 1.1.( 2) 3.2.2 1.1.2 0 t t t mt

 

       



, 0 2 1 1 3.2.1 1.1.( 1) 3.2.1 1.1.( 1) 10 0 t t t mt

 

        



. 1 1 1 1 1 1 1 3.( 2)( 2) 1.( 2)( 2) 3.2.2 1.2.2 32 2 3 t t t mt nw

 

            



, 3 3 1 3 1 1 1 3.1,1 1.2.2 3.( 1)( 1) 1.( 2).( 2) 14 1 1 t t t mt nw

 

            _ 



. □ Example 5

Take

n



6

,

q



5

,





2

, hence

k



4

and

kn



24

. We choose as primitive 24th root of unity again a zero of

x

2

 

x

2

, which yields the following

indexing of the irreducible polynomials contained in

x

24



1

:

(1) 0 ( ) 1 P x  x , P₄(1)( )x x2  x 1, P₈(1)( )x x2  x 1, P₁₂(1)( )x  x 1, (2) 2 1 ( ) 2 P x x  x , P₉(2)( )x x22, P₁₃(2)( )x x2 x 2, (3) 2 3 ( ) 2 P x x  , P₇(3)( )x x2 2x2, P₁₉(3)( )x x2 2x2, (4) 2 2 ( ) 2 1 P x x  x , P₆(4)( )x  x 2, P₁₄(4)( )x x22x1, P₁₈(4)( )x  x 2. As for the set

T

2, we define

T

2



{1,9,13}

.

Next, we shall compute the primitive idempotent polynomials

15

1

s



,

m

₁



2

,

a

₁



1

, 2 2 1 1 1 1 1 2 1

2 1

3

2

2 2

m p

m



 

 





,

2

s



,

m

₂



2

,

a

₂



2

, 2 4 1 1 2 2 2 4 2

2 2

3

2

4 2

m p

m



 

 





, 2 2 4 5 1 ( )x 2 2x 2x x 2x



     .

The equations



₁2( )x  1



₁2( )x ,



₁2( )x (6.2)1xh x g x'( ) ( ),

g x

( )



x

2

 

x

2

and

4 3 2

( )

2

1

h x



x

  

x

x

x



give the same result. Similarly, we obtain:

9

t



, 2 1 9 9 0 0 0 1 0

2 1

2

2

1 1

m p

m



 

 





, 2 2 9 9 9 1 1 2 9

2 0

0

2

2 2

m p

m



 

 



, 2 4 9 9 18 2 2 4 18

2 2

1

2

4 1

m p

m



 

 

 

, 2 2 4 9( )x 2 x 2x



   .

13

t



, 2 1 13 13 0 0 0 1 0

2 1

2

2

1 1

m

p

m



 

 





, 2 2 13 13 13 1 1 2 13

2 1

2

2

2 2

m

p

m



 

 



, 2 4 13 13 2 2 2 4 2

2 2

2

2

4 2

m

p

m



 

 



 

,



₁₃2( )x  2 2x2x2x42x5.

We verified the relation



i2( )x 2 



i2( )x for all three values of i.

Finally, we present the three primitive idempotents as the colunns of a

3 3



-marix: t 1 9 13 s

m

_s

w

_s



2 2 2 2 0 2 2 1 2   _         

0

1

2

1

2

2

1

4

3

One can easily check that all relations of Theorem (iv) and (v) are satisfied in this case, e.g.

3 3 2 3 1 1 1 3.( 2).( 2) 3.( 1).( 1) 3.( 2).( 2) 27 2 3 t t t mt nw

 

            



. □

In order to put the concept of the matrix with orthogonal columns and rows of the previous examples in a general form, we now define

_, : as t s / s

s t n s m pt st mst

  





 

  ,

s



S

,

t



T

, (8)

16

Definition 3

The matrix Mn q, ,, shortly M , is defined as the

n

₀



n

₀-matrix with elements _{s t, s}t

t n M m



 ,

(

/

as

)

s t

m

t 







and the



-adjoint matrix

M

* as the n₀n₀-matrix with elements

M

_{t s}*_,



w

_s



_t_s, for all

s



S

and

t



T

.

Theorem 4

The matrix

M

has the following properties.

(i) The matrix

M

* is the inverse of the matrix

M

, i.e. u s*, s t, u t,

s S

M M











and s t, t r*, s r, t T

M M











.

(ii) If e x( )Cn q, , is the idempotent generator of the code

1 2 ( ) ( ) ( )

( )

( )....

( )

l i i i

P



x P



x

P



x





and if one writes , ,

( )

( )

n q s s s S

e x

c x

 

_







, then the coefficients



_s are uniquely determined by the matrix equation

M

 



where



is the column vector of length n_o with zeros on the positions i₁, i₂,...., i_l and ones

elsewhere, while



is the column vector 0

0 1 2

( ,

,....,

)

T

( )

n n

GF q

 





. Proof

(i) This is nothing else as the matrix form of the orthogonality properties of Theorem 1 (v) and (vi). By applying these properties, we derive _{u s}*_{, s t, s} u_{s s}t

s s t n M M w m



_









, t u s s s t u s t

n

w

m



 











and * , , , 1 t t t t s t t r s r r r s r s r t t t t t n M M w nw m

 

 m

 





  







. (ii) The columns of

M

are the representation vectors



_stof the primitive idempotents



_t( )x for all

, ,

n q

t



T

. So, when taking

 

: _t (0,....,1,...., 0) with all elements equal to 0 except the element on position

t

we obtain precisely the vector



st. The statement in (ii) now follows immediately. □

Remarks

As for the notation in Theorem 3 and in its proof, one should keep in mind that the sets S and

T

need not always be identical, and even that it is not always possible to choose their elements such that they become identical. E.g. if sS and sT for some value of s, one has to take for

t

in



_{s t,} the

integer t'T with tC_tn q_', . The conclusion in the above proof that the matrix

nM

* is a right inverse of

M

can also be drawn

from Theorem 1 (iv). As for the variables



_{s t,} introduced in (8), we can write



_{s t,}  g p_st _sts, where

: / s

t

s t st

g m m is equal to the quotient of the sizes of the constacyclotomic cosets C_t and C_sts, or equivalently of the degrees of the irreducible polynomial

P

_t( )

( )

x

and of P_st(s)( )x which has as zeros the s-powers of the zeros of P_t( ) ( )x . Consequently, the value of



_{s t,} is equal to the sum of the s

-powers of the zeros of

P

_t( )

( )

x

. In general we have _, as t

s t

n

s

17 m_s



_{s t,} m_t



_{t s,} , [4, Th. 11 (i)] _{s r, r t, s, t} r T

n



 



_ 





, [4, Th. 11 (ii)] s s t, s u, t t, u s S

m

nm



 



_ 





, [4, Th. 11(iii)] t 1 s t, r t, s 1 s, r t T

m

nm



 



   





. [4, Th. 11 (iv)] Actually, C_tn q, ,1C_tn q_,₁ which is not identical to C_tn q, , but since this identity holds for all tT we are

entitled to replace (m1_t)1 by m_t1 in the derivations we made and also in Definition 3. Notice that in the case



1 the sets

S

 and

T

 can be chosen identical. The last two relations are special forms of the general orthogonality relations in terms of the



-variables as follows from Theorem 1 (v) and (vi) 1 _{, , ,} s s s t s u t u a s S t m n m  

 





  



, 1 _{, , ,} s s s t r t s r a t T t m n m  

 





  



.

As an alternative for the matrix

M

in Definition 3, we now introduce the matrix



with entries

,

:

,

s t



s t

 

. According to the above formulas,



is an orthogonal matrix with weight factors

s s a m



and 1 t

m for the orthogonality of respectively its rows and its columns.

The orthogonality relations for the primitive idempotents represented by the column vectors



t,

t



T

, with components



_st,

s



S

, or equivalently, by the column vectors



t, show a certain similarity with the orthogonality relations of the characters



_sjof the irreducible representations

( j

D G) of a finite group G. Within this context cyclotonomials c_s( )x are to be considered as the counterparts of the classes C_s of conjugated elements in the group. The irreducible polynomials

( )

( ) t

P x ,

t



T

, which take care of the labelling of the idempotents, seem to have no obvious counterpart in character theory. Usually, an irreducible character (and the corresponding matrix representation) is indicated by some ad hoc parameter

j

or by its dimension



j

(1)

. Since

1

0,

(

0 0

)

t

t

g p



 

is equal to the sum of the zero-powers of the zeros of P_t( ) ( )x , we may conclude that

0,t

m

t







. So, the row of



with label 0 contains the ‘sizes’ m_t of the irreducible polynomials, just like the row in a character table labelled by 1 (from the class C₁ (1)) contains the ‘sizes’ of the

irreducible representations



j(1). For



1 the resemblance between idempotent tables and character tables is even stronger. Since

0 1

,0

:

0

s

g p

s



 

is equal to the sum of the s-powers of the zeros of

P

₀(1)

( )

x

 

x

1

,



_s,0 1 for all

s



S

. So, the idempotent column vector



0 can be seen as the counterpart of the trivial character

1



, in the literature also denoted by 1_G. Furthermore, a cyclonomial is self adjoint if and only if

, ,

( ) ( )

n q n q

s n s

18

definition of adjoint irreducible polynomial it follows immediately that Pt(1)( )x is self adjoint if and

only if C_tn q, C_{n t}n q_, . For



1, the index sets

S

n q, ,1 and

T

n q, ,1 can be chosen identical. So,in the case

1



 , the number of self adjoint cyclonomials and the number of self adjoint irreducible polynomials, and hence also the number of self adjoint primitive idempotent generators, are equal for fixed values of nand

q

. This equality resembles the property that the number of real characters, i.e. characters of self adjoint or self conjugated irreducible representations over ₵. is equal to the number of self inverse classes of conjugated elements in the group. We can also prove this property by exploiting the orthogonality relations like is done in character theory. From the first orthogonalty relation for



1

it follows that a column



t is self adjoint if and only if 1 _{s s t, s t,} 1 s

t m

nm



 

 , while the rhs is equal to 0 if



t is not self adjoint. Hence, the total number of self adjoint columns of



is equal to

, , , 1 s s t s t s t t m

n



m

 

. Similarly, the second orthogonality relation gives that a row



s of



is self adjoint if and only if s 1 _{. ,} 1

s t s t

t t

m

n



m

 

 , and if not then the rhs equals 0. Hence, the total number of self adjoint rows of



also equals , ,

, 1 _s s t s t s t t m n



m

 

.

3. Conjugated constacyclotomic cosets

In this section we shall introduce the notion of conjugated constacyclotomic coset with respect to certain transformations. Unlike as in [1-4], in this report we shall consider cyclotomic and

constacyclotomic cosets as ordered sets. Furthermore, we study sets of integers mod n of the form

, ,

n q t

aC





r

. It appears that for special values of r, such a set is again a constacyclotomic coset, which of course has to be labelled as C_{at r}n q, ,_, since atr is an element of it. In the next we shall describe this process as the mapping C_tn q, , C_{at r}_ induced by aC_tn q, , r, or shortly by

, , , ,

n q n q

t at r

C  C _. Like on earlier occasions, we also shall drop the parameters n and

q

.

Theorem 5 Let _{0 1} 1 ( , ,...., ) t t _m

C  c c c _ ,

c

₀



t

, be some constacyclotomic coset, and let the notation aC_t r

stand for the ordered set _{0 1}

1

( , ,...., )

t m

ac r ac r ac _ r . (i) If a is a fixed integer and if there is an integer

r

₀ such that

a

 

1

kr

₀



0

mod

n

/ ( , )

l n

, then

there are

( , )

l n

elements

kr



Z

_n such that

a

  

1

kr

0

mod

n

/ ( , )

l n

, and they can be written as

0

/ ( , )

kr



kr



jn

l n

,

0

 

j

( , )

l n

. For all r- values satisfying the above relation one has aC_tr

*

t at r at

m

C

m

   



, where the factor t

at

m

m



 denotes the number of times that each element of Cat r 

 occurs in

the right hand side.

(ii) For any

r



jn

/ (

q



1, )

n

,

0

  

j

(

q

1, )

n

, the mapping C_t C_{t r} defines a permutation on

the family of constacyclotomic cosets

{

C

_t∣

t



T



}

.