Primitive idempotent tables of cyclic and constacyclic codes

For any λ∈GF(q)*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lambda \in GF(q)^{\ast} $$\end{document} a λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lambda $$\end{document}-constacyclic code Cn,q,λ:=⟨g(x)⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C^{n,q,\lambda } : = \,\langle g(x) \rangle $$\end{document}, of length n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n $$\end{document} is a set of polynomials in the ring GF(q)[x]/xn-λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ GF(q)[x]/x^{n} - \lambda $$\end{document}, which is generated by some polynomial divisor g(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ g(x) $$\end{document} of xn-λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x^{n} - \lambda $$\end{document}. In this paper a general expression is presented for the uniquely determined idempotent generator of such a code. In particular, if g(x):=(xn-λ)/Ptn,q,λ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ g(x): = (x^{n} - \lambda) / P_{t}^{n,q,\lambda } (x) $$\end{document}, where Ptn,q,λ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P_{t}^{n,q,\lambda } (x) $$\end{document} is an irreducible factor polynomial of xn-λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x^{n} - \lambda $$\end{document}, one obtains a so-called minimal or irreducible constacyclic code. The idempotent generator of a minimal code is called a primitive idempotent generating polynomial or, shortly, a primitive idempotent. It is proven that for any triple (n,q,λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (n,q,\lambda ) $$\end{document} with (n,q)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (n,q) = 1 $$\end{document} the set of primitive idempotents gives rise to an orthogonal matrix. This matrix is closely related to a table which shows some resemblance with irreducible character tables of finite groups. The cases λ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lambda = 1 $$\end{document} (cyclic codes) and λ=-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lambda = - 1 $$\end{document} (negacyclic codes), which show this resemblance most clearly, are studied in more detail. All results in this paper are extensions and generalizations of those in van Zanten (Des Codes Cryptogr 75:315–334, 2015).

so-called negacyclic codes [2,8,26]. For general values of λ ( 0), constacyclic codes have first been introduced in [2]. For more general information about these codes, we refer to [1, 3-5, 20, 21] and to the lists of references in these publications.
Let the decomposition of x n − λ into monic irreducible polynomials over G F(q) be given by where T n,q,λ is an index set containing the indices of all these irreducible polynomials. If f t (x) : (x n − λ)/P n,q,λ t (x) for some fixed t ∈ T n,q,λ , then the code f t (x) is called a minimal or irreducible constacyclic code. In algebraic terms, such a code is a minimal ideal in the ring R n,q,λ . The code P n,q,λ t (x) is called a maximal constacyclic code. An idempotent polynomial in R n,q,λ is a polynomial e n,q,λ (x) ∈ R n,q,λ with the property that e n,q,λ (x) 2 e n,q,λ (x). (2) It will be clear that if (2) holds, then all positive powers of e n,q,λ (x) are identical. If e n,q,λ (x) generates the code C, then it is called an idempotent generating polynomial of C, or shortly an idempotent generator. One can easily prove that each constacyclic code has a uniquely determined idempotent generator (cf. [18,19,24]). The idempotent generators of minimal constacyclic codes are denoted by θ t (x) and those of the maximal constacyclic codes by ϑ t (x), t ∈ T n,q,λ [16,22,25]. The polynomials θ t (x) are often called primitive idempotent polynomials, since any idempotent generator can be written as a linear combination of these polynomials for fixed values of n, q and λ. Constacyclic codes with special parameter values or constacyclic codes constructed by special methods are discussed in [9,11,12,14,15].
In the next section we shall formulate a few simple properties of (primitive) idempotent generating polynomials which are well known for cyclic codes, and which also hold for constacyclic codes. The proofs are completely similar to those for cyclic codes, and will therefore be omitted in most of the cases. Actually, all relations mentioned in Sect. 2 can be seen as special cases of properties of idempotents in the context of semi-simple algebras (cf. [6,23,24]).
The notation C n,q,λ stands for a λ-constacyclic code of length n over G F(q), where the positive integer n, the prime power q and the parameter λ satisfy the conditions (n, q) 1, λ ∈ G F(q) * . ( Under these assumptions x n −λ has no multiple zeros, and hence the irreducible polynomials have no common zeros. Throughout the paper we shall assume that (3) holds, without stating so every time. In Sect. 2 we also present a general formula which enables us to determine the idempotent generator for any constacyclic code C n,q,λ , where the three parameters n, q and λ satisfy the conditions in (3). In Sect. 3 we discuss codes C n,q,λ for fixed values of n and q and for various values of λ as subcodes of the cyclic code C kn,q,1 , where k is the multiplicative order of λ in G F(q). In Sect. 4.1 the notion of constacyclotomic coset is introduced as a generalization of cyclotomic coset, and in Sect. 4.2 the notion of constacyclonomial, generalizing cyclonomials. The vector space spanned by these constacyclonomials for fixed n, q and λ is called A n,q,λ . Furthermore, this vector space is equipped with a bilinear form. In Sect. 5 it is shown that with respect to this bilinear form, both the constacyclonomials and the primitive generator polynomials constitute an orthogonal basis of A n,q,λ . The orthogonal transformation matrix between these two bases can be interpreted as an orthogonal table of primitive idempotent generators. It turns out that such tables resemble, in a way, the wellknown irreducible character tables of finite groups, thus generalizing similar results in [25]. Therefore, we shall speak of primitive idempotent tables. The most striking examples of this resemblance are obtained by taking λ 1 (cyclic codes) or λ −1 (negacyclic codes). Respectively in Sects. 6 and 7 these cases are discussed in more detail. Among other things, we define the notions of r -conjugateness for primitive idempotent generators and blocks of r -conjugated idempotents. As for our notation in the remaining sections, if this will not give rise to confusion we shall drop the indices n and q from the names of variables for reasons of convenience. So, we shall write e λ (x) instead of e n,q,λ (x), P λ t (x) for P n,q,λ t (x), etc. Only in places where the variable n takes on various values such as in Sect. 4.2, we shall use the more extended notation. In order to keep the reader aware of the dependence on n and q, we always maintain the full notation in the names of the sets these variables are taken from, like R n,q,λ , A n,q,λ , S n,q,λ , T n,q,λ , C n,q,λ and C n,q,λ t .

Idempotent generators
In order to formulate the announced properties, we introduce a couple of notions and corresponding notation. Firstly, we write the n zeros of x n − λ as αζ i , i ∈ {0, 1, . . . , n − 1}, where ζ is a primitive nth root of unity in some extension field of G F(q) and α a fixed element of the same extension field, say in F : G F(q)(α, ζ ), which satisfies α n λ (cf. also Theorem 4). From standard theory on polynomials in G F(q) [x], we know that αζ i and αζ j : (αζ i ) q are zeros of the same irreducible polynomial, for any i ∈ {0, 1, . . . , n − 1}. As a consequence, when having chosen a fixed element α ∈ F, one can take for the index set T n,q,λ in (1) an appropriate subset of { 0, 1, . . . , n − 1}. Usually, we shall take the minimal i-value for which αζ i is a zero of the irreducible polynomial to be indexed. In case that λ 1 we can take α 1, and we obtain the usual set of indices representing the cyclotomic cosets modulo n with respect to q, called the q-cyclotomic cosets modulo n.

Theorem 1 Let C :
g(x) be a λ -constacyclic code in R n,q,λ and let h(x) , defined by g(x)h(x) x n − λ , be its check polynomial.
(i) If e λ (x) is the uniquely determined idempotent generator of C , then there exist polynomials p(x) and q(x) such that e λ (x) p(x)g(x) and g(x) q(x)e λ (x) in R n,q,λ .
(v) If C 1 and C 2 are λ -constacyclic codes with idempotent generators e λ 1 (x) and e λ 2 (x) , then C 1 ∩ C 2 and C 1 + C 2 are also λ -constacyclic codes with idempotent generators e λ 1 (x)e λ 2 (x) and e λ 1 (x) + e λ 2 (x) − e λ 1 (x)e λ 2 (x), respectively. The proofs are completely similar to the proofs for cyclic codes which can be found e.g. in [18,19,24]. The same holds for the properties listed in the next theorem.
Theorem 2 Let {θ t (x) t ∈ T n,q,λ } be the set of primitive idempotent generators of the λ -constacyclic codes generated by divisors of x n − λ . Then one has for all t, u ∈ T n,q,λ the following properties: is the idempotent generator of some λ -constacyclic code in R n,q,λ , then there exist elements ξ 1 , ξ 2 ,…., ξ r ∈ G F(q) such that e λ (x) , t * ∈ T n,q,λ −1 , such that the zeros of P λ −1 t * (x) are the inverses of the zeros of P λ t (x) . The corresponding primitive idempotent generator satisfies θ t * (x) : λx n θ t (1/x).

Proof
The proofs for (i)-(v) are straightforward and similar to the proofs for cyclic codes, i.e. for λ 1.
(vi) That P λ t * (x) is a monic irreducible polynomial in R n,q,λ follows immediately from its definition (cf. also [17]). We know that x n − λ q(x)P λ t (x) for some polynomial q(x) ∈ R n,q,λ . From this equality we derive 1 − λx n x n−m λ , and it has value λ · 0 0 when substituting one of the other zeros of We shall present a slightly different proof for part (iv), right after the next theorem which provides us with a simple expression for the uniquely determined idempotent generator of a constacyclic code.
is a divisor of x n − λ in R n,q,λ , with (n, q) 1 , then the idempotent generator of the λ -constacyclic code g(x) is given by the polynomial Proof Because of the assumption (n, q) 1 the polynomials g(x) and h(x) have no common zeros, and since both are monic we have (g(x), h(x)) 1. Hence, there exist polynomials a(x) To determine a(x), we take derivatives of both sides of the relation Hence, a(x) (nλ) −1 xh (x) and the relation in (i) now follows, as well as the relation in (ii) by interchanging g(x) and h(x).
We remark that the expression for e λ (x) in Theorem 3 generalizes the expression for the idempotent generator of a cyclic code in [25] which on its turn was a generalization of the special case of binary cyclic codes (cf. [18,19,24]). As an application of Theorem 3 (i), we now present an alternative proof for Theorem 2 (iv).

Example 4
Consider the primitive idempotent polynomials θ t (x), t ∈ T n,q,λ , belonging to the polynomial in (1) , according to Theorem 3 (i). Applying the rule for determining the derivative of a product of functions yields t∈T n,q,λ θ t (x) (nλ) −1 3 Constacyclic codes C n,q, for various values of (i) If the n 0 : T n,q,λ irreducible factor polynomials P λ have order e t , 1 ≤ t ≤ n 0 , then e is equal to the least common multiple e 1 , e 2 , . . . , e n 0 . (ii) The order e of x n − λ is equal to kn, and if α is a zero of x n − λ of order e, then all its zeros can be written as αζ i , 0 ≤ i < n, where ζ : α k . Furthermore, one has (iii) If α is some zero of x n − λ and if there is no integer i, 0 < i < n, with α i ∈ G F(q), then α has order kn in F. Conversely, if α i μ ∈ G F(q), 0 < i < n, there is a minimal divisor d of n, d < n, such that the order of α is hd where h is the order of μ in G F(q). (iv) The order of α is equal to kn if and only if n is a divisor of hd. If n is not a divisor of hd, then α is a zero of x d − μ, which is a factor of x n − λ.

Proof
(i) The irreducible polynomial factors of x n − λ are pairwise prime to each other, since (n, q) 1, and hence x n − λ has no multiple zeros. So, the statement is an immediate consequence of [17,Theorem 3.9]. (ii) Let G be the multiplicative group consisting of the e zeros of x e − 1 in some extension field of F and let β be a generator of this group. Since x n − λ is a divisor of x e − 1, the group G contains n different elements β b satisfying β bn λ. It follows that there are n different elements β b−b 0 all satisfying β (b−b 0 )n 1, for some fixed integer b 0 , and so these elements form a subgroup of order n. Hence, n is a divisor of e. Furthermore, λ e/n β bn e/n β be 1, and hence e akn for some positive integer a. However, since α kn λ k 1 for any zero α of x n − λ, we have that e ≤ kn. So, a 1 and e kn. If we define ζ : β k , it follows that ζ n (β e/n ) n 1 , and so ζ is a primitive nth root of unity, since n is minimal positive with respect to this property. Hence, all zeros of x n − λ can be written as β 1+ik , 0 ≤ i ≤ n − 1. Defining α : β yields the first equality in (ii). The second equality now follows easily by applying that α n λ implies (α j ) n λ j and from the fact that all zeros of the polynomials x n − λ j , 0 ≤ j ≤ k − 1, are different. (iii) Let the order of α in F be equal to f . Then we have f ≤ kn. We also have f ≥ n, because of the condition on α. Hence, we can write f sn + t, with s ≥ 1, 0 ≤ t < n. It follows that α sn+t λ s α t 1, and so α t λ −s ∈ G F(q). Because of the condition on α, this can only be true for t 0, and λ s 1. Therefore, s ≥ k and f ≥ kn. We conclude that f kn. Conversely, assume α i ∈ G F(q) for some i with 0 < i < n. Then we have for all integer values a and b that α an+bi ∈ G F(q), in particular for those values a and b for which an + bi (n, j). So, for all i satisfying the above assumption, we have α (n,i) ∈ G F(q). Let d be the greatest common divisor of these i-values, then α d μ and d is minimal with respect to this property. Similarly to the proof of the first part it now follows, replacing n by d and k by h, that the order of α is equal to hd. We remark that the proof of Theorem 5 (iii) is based on the proof of Lemma 3.17 in [17] which deals with a similar property for the order of an arbitrary polynomial f (x) ∈ G F(q) [x], f (0) 0, of positive degree. We also notice that (e, q) (kn, q) 1, due to (3) and the fact that k is a divisor of q − 1, and so x e − 1 has no multiple zeros. The following corollary is based on the fact that if α is a zero of x n − λ, then α j is a zero of x n − λ j for 0 ≤ j ≤ k − 1. Together with Theorem 5 (ii) this yields the following result.

Corollary 6
Let j be some integer with 0 ≤ j ≤ k − 1. If H : ζ , ζ : α k , with α a zero of x n − λ of order kn, is the uniquely determined subgroup of G : α of order n, then the cosets of H in G are H j α j H and H j consists of all n zeros α j ζ i , 0 ≤ i ≤ n − 1, of the polynomial x n − λ j .

Theorem 7
Let g(x) be a polynomial dividing the polynomial x n − λ of order e( kn). If e λ (x) is the idempotent generator of the λ-constacyclic code g(x) λ in R n,q,λ and e(x) the idempotent generator of the cyclic code g(x) in R e,q (: R e,q,1 ), then e λ (x) e(x) mod x n − λ.
Proof If h λ (x) and h(x) denote the check polynomials of g(x) in R n,q,λ and in R e,q respectively, we can write ( . Taking derivatives on both sides of this equality, applying Theorem 3 (i) and dividing by x n − λ, yields the rela-

with the polynomial t(x)
: . Since x n − λ divides x e − 1, the above equality also holds modulo x n − λ. Substituting x n λ and x e 1 then gives modulo x n − λ that t(x) k/λ and next e λ (x) e(x).

Generalization of cyclotomic cosets and cyclonomials
From standard results on cyclotomic cosets (cf. [17]) it is well known that the zeros of any irreducible factor of x n − 1 can be written as ζ t , ζ tq , . . . , ζ tq m t −1 for some integer t, where ζ is a primitive n th root of unity in some extension field of G F(q), while m t is the degree of that polynomial. So, by taking these integers t as elements of the index set T n,q (: T n,q,1 ), we establish a one-one correspondence between the irreducible polynomials P 1 t (x) and the q-cyclotomic cosets mod n C n,q t (t, tq, . . . , tq m t −1 ), (4) where m t is the smallest positive integer which satisfies t(q m t − 1) 0 mod n (cf. also [25]). In the next subsection we shall generalize this correspondence for those irreducible polynomials which play a role in constacyclotomic cases, i.e. when they are divisors of x n −λ, λ 1.

Definition 7
For any triple of parameters n, q and λ satisfying condition (3), the set where m λ t is the smallest positive integer satisfying c m λ t −1 q + l c 0 , is called a (q-) constacyclotomic coset modulo n.
Next, we shall derive a number of properties of constacyclotomic cosets which we shall need in the remaining sections. In the formulation of these properties we shall use the notation aC n,q,λ t +b : aC n,q,λ t Theorem 8 Let q be a prime power, n an integer with (n, q) 1 and let λ ∈ G F(q) * have order k. Let furthermore α be a zero of x n − λ ∈ G F(q)[x] of order kn and let ζ : α k .
(i) The zeros of some irreducible polynomial P λ t (x) over G F(q) contained in x n − λ can be written as αζ c where c runs through the set C n,q,λ t , while m λ t is equal to the degree of that polynomial.
is equal to the smallest positive integer which satisfies the relation , where the notation a × C n,q,λ or a × C n,q means that each integer of the relevant coset occurs a times in the multiset at the left hand side of the equality.

Proof
(i) Let α be a zero of the irreducible factor P λ (x) of x n − λ of degree m 0 . Then we can From Theorem 5 we know that for all relevant i, we can write α q i αζ c i for some integer c i ∈ {0, 1, . . . , n − 1}. Hence, α q i+1 α q ζ qc i αα q−1 ζ qc i αζ l+qc i by using q − 1 kl, and we obtain (4) and (5) with t t 0 0 define P λ t 0 (x) : P λ (x). Next, let t 1 be the least integer in {0, 1, . . . , n − 1}\C n,q,λ 0 and define P λ t 1 (x) of degree m t 1 as the irreducible factor of x n − λ which has αζ t 1 as zero. Similarly as before it appears that this polynomial is defined by (4) and (5) (v) These relations follow immediately from (ii).

Remark 9
We remark that putting λ 1, and hence k 1, l q − 1 in (5) and (6), does not provide us with the cyclotomic cosets (4). In terms of the integers of C n,q,1 t (c 0 ( t), c 1 , . . . , c m t −1 ), the zeros of the corresponding irreducible polynomial can be written as αζ c 0 , αζ c i , …, αζ c m t −1 , with α ζ as primitive n th root of unity. We call this the α, ζ -representation. On the other hand, the integers of (3) give these zeros in the form ζ t , ζ tq , . . . ζ tq m t −1 , the ζ -representation. Application of Theorem 8 (v) with λ 1 and k 1, shows that the two types of cosets are related by C n,q,1 This relation implies that m 1 t−1 m t for all t ∈ T n,q . In the next we keep calling C n,q,1 t , t ∈ T n,q,1 , a constacyclotomic coset and C n,q t , t ∈ T n,q , a cyclotomic coset. Furthermore, as was already remarked in Theorem 8 (v), the set kC n,q,λ t + 1 is, strictly speaking, an ordered multiset such that any integer it contains occurs the same number of times. This is due to the fact that all operations on the integers have to be carried out modulo n. Finally, we emphasize that the third relation in Theorem 8 (v) does not always define a one-one mapping from the set of constacyclotomic cosets to the set of cyclotomic cosets for λ 1. E.g. 4C 14,5,2 0 + 1 (1,5,11,13,9,3) and 4C 14,5,2 4 + 1 (3,1,5,11,13,9).
Next, we present a theorem which shows how to determine constacyclotomic cosets C n,q,μ t for various μ ∈ G F(q) * in a way, other than by the recurrence relation (6) or by the rules of Theorem 8 (ii). To this end we shall need the irreducible polynomial which has as zeros the s-powers of the zeros of P λ t (x) (cf. Theorem 8), for t ∈ T n,q,λ and for s ≥ 0. This polynomial is an irreducible factor of x n − λ s , denoted by P λ s t s (x). (iv) For any s, 1 ≤ s ≤ k, the mapping of {0, 1, . . . , n − 1} into {0, 1, . . . , kn} defined by i → ki + s yields a one-one correspondence between the constacyclotomic cosets C n,q,λ s t , t ∈ T n,q,λ s , and the cyclotomic cosets C kn,q kt+s , kt + s ∈ T kn,q . For s 0 the mapping yields a one-one correspondence between the cyclotomic cosets C n,q t , t ∈ T n,q , and the cyclotomic cosets C kn,q kt , kt ∈ T kn,q .
(v) For any s ≥ 0 the s-powers of the zeros The definition of the order of a polynomial implies that x n − λ |x kn − 1. So all zeros of x n − λ lie in an extension field F of G F(q). For any s, 1 ≤ s ≤ k, we have that the order of λ s is a divisor of k, and so the order of x n − λ s divides kn as well. Hence, all kn zeros of these polynomials are in F. Now, (n, q) 1, and since k |q − 1 we also have (kn, q) 1, which implies that the polynomials have no zeros in common. This proves the factorization. (ii) This follows immediately from Theorem 8 (i) and from the relation (α s ) n λ s .
be some cyclotomic coset. Since 0 ≤ a ≤ kn − 1, there is precisely one way to write a kt + s, for any s with 1 ≤ s ≤ k. It follows that 0 ≤ t ≤ n − 1, and so there is precisely one constacyclotomic coset C n,q,λ s t which is mapped to C kn,q a . If s 0 the zeros of x kn − 1 are written as α kt , and hence the zeros of x n − λ 0 x n − 1 as α t , 0 ≤ t ≤ n − 1. This proves the second statement in (iv) (cf. also Remark 9).
(v) If αζ i is a zero of x n − λ, then α s ζ is is a zero of x n − λ s . Let αζ i be a zero of P λ t (x), then α s ζ is is a zero of some irreducible polynomial P λ s t s (x), and this polynomial is the same for all i ∈ C n,q,λ t . We put α : α s and ζ : α k/(k,s) , where k/(k, s) is the order of λ s in G F(q), and so ζ α sk/(k,s) ζ s/(k,s) . It follows that the zeros of P λ s t s (x) can be written as α ζ j , j ∈ C n,q,λ s t s , with t s (k, s)t. Each integer in C n,q,λ s t s corresponds m λ t /m λ s t s times to some integer in C n,q,λ t .

Example 11
Take n 8, q 5 and λ 2. It follows that k : ord 5 (2) 4. Since x 8 − 2 does not divide x 16 − 1, its order is kn 32. We have the following factorization  (24). The factorization of the polynomials x 8 −λ s , s 0 and s 2, into irreducible polynomials over G F (5) is respectively Proof (i) We know from (6) that the elements of C n,q,λ t satisfy c i+1 qc i + l mod n for 0 ≤ i ≤ m λ t −1. Now, we define d i : rc i +a mod n. If we require that the elements of C n,q,λ t keep their mutual order under the mapping on C λ rt+a , we must have that d i+1 rc i+1 + a mod n. Consequently, d i+1 − qd i − l rc i+1 + a − rqc i − aq − l (r − 1)l − a(q − 1) 0 mod n, and the condition on a follows by applying q − 1 kl. Since (r , n) 1, multiplying the integers c i of C n,q,λ t by r does not alter the size of the coset, and neither does adding the same integer a to all rc i mod n.
(ii) and (iii) follow immediately from (i) by substituting respectively r 1 and r −1.
We emphasize that the conditions on r are sufficient but not necessary for the properties mentioned in Theorem 12. As the proof in (i) shows, they are necessary as well if one requires that the mutual order of the integers in C n,q,λ t is not to be changed by the mapping. An example is provided by the constacyclotomic cosets C 12,7,2 0 (0, 2, 4, 6, 8, 10), C 12,7,2 1 (1,9,5) and C 12,7,2 3 (3, 11, 7), with k 3 and l 2. The equation 3a + 2 0 mod 6 has no solutions, but the mapping t → −t + 2 defines a permutation of order 2 on the set of the three constacyclotomic cosets, while it reverses the order of the integers. As preparation for Sect. 6 and 7, we notice that for λ 1 and for λ −1 an integer a as mentioned in Theorem 12 (iii) exists. In the cyclic case of λ 1, we have k 1 and hence a −2 is a solution of the equation in (iii). So, the mapping t → −t −2 mod n yields a permutation of order 1 or 2 on the set C n,q,1 t t ∈ T n,q,1 . By applying (7), one can see that this is equivalent to a permutation on the set C n,q t t ∈ T n,q induced by t → n −t. In the negacyclic case of λ −1, we have k 2 which provides us with a −1 and the mapping t → n −t −1 which acts similarly on C n,q,−1 t t ∈ T n,q,−1 . We define C

Constacyclonomials c n,q, s (x)
A second notion in the theory of cyclic codes that we shall generalize is that of cyclonomic polynomial or cyclonomial (cf. e.g. [25]). To each cyclotomic coset C n,q s of size m s there corresponds a cyclonomial c n,q s (x) : x s + x sq + · · · + x sq ms −1 mod x n − 1.
Clearly, such a polynomial, shortly written as c s (x), (cf. Sect. 1) has the property In the following definition is s an integer of {0, 1, . . . , n − 1}, and λ an arbitrary element of G F(q) * .

Definition 13
The polynomial c λ s (x) : x s + x sq + · · · + x sq m λ s −1 mod x n − λ in R n,q,λ is called a monic constacyclonomial of size m λ s , if it is not the zero polynomial and if m λ s is the smallest positive integer such that x sq m λ s −1 q x s mod x n − λ.
Since β q β, for any β ∈ G F(q) * , we could call any polynomial βc λ s (x) with c λ s (x) satisfying the equality in Definition 13, a constacyclonomial. However, we shall reserve this term for monic polynomials. For λ 1 we obtain the usual cyclonomials. We identify these two types of cyclonomials by writing c 1 s (x) ≡ c s (x). It will be obvious that if c λ s (x) contains a term βx t , β ∈ G F(q) * , then c λ t (x) β −1 c λ s (x), and so c λ t (x) and c λ s (x) are linearly dependent polynomials. For fixed values of n and q, we shall use the notation S n,q,λ for a maximal set of indices of independent constacyclonomials. Usually, we take the lowest exponent of the xpowers as index of a constacyclonomial, similarly as in the case of constacyclotomic cosets, but actually one can take any of its exponents because of the above mentioned dependency. Let s ∈ S n,q,λ and assume that c λ s (x) does not contain a term βx n−s . Then it follows easily from Definition 13 that c λ n−s (x) is a different constacyclonomial of the same size. The monic constacyclonomials c λ s (x) and c λ n−s (x) are called a pair of conjugated constacyclonomials. If c λ s (x) does contain such a term βx n−s , it is called a self conjugated constacyclonomial. If c λ s (x), s ∈ S n,q,λ , is not self conjugated, we assume that n − s is also in S n,q,λ , even if it is not the lowest exponent in the relevant polynomial.

Definition 14
The conjugate c λ * s (x) of the constacyclonomial c λ s (x), s ∈ S n,q,λ , is defined as From the condition prior to its definition, it follows that c λ * s (x) is a (monic) constacyclonomial for all s ∈ S n,q,λ . Next, we define the following subset of R n,q,λ spanned by the constacyclonomials with fixed values for n, q and λ A n,q,λ : This set A n,q,λ and its elements have the following simple properties.

Theorem 15 (i) The polynomial c λ s (x)is a constacyclonomial of size m λ s if and only if it is not the zero polynomial and if m λ s is the smallest positive integer satisfying s(q m λ s −1) 0 mod kn. (ii) Any polynomial p(x) of A n,q,λ satisfies p(x) q p(x). (iii) Let m be the smallest positive integer such that x sq m
βx s mod x n − λ for some β ∈ G F(q) * . Then the polynomial p(x) x s + x sq + · · · + x sq lm−1 , l : or d q (β), is the constacyclonomial c λ s (x) of size m s m for β 1, whereas p(x)is the zeropolynomial for β 1.

(iv) A constacyclonomial has no proper subpolynomial which is also a constacyclonomial. (v) If all nonzero coefficients of c λ
s (x) are changed into 1, one obtains the cyclonomial c s (x). When writing s(q m λ s − 1) an + b, with a ≥ 0, 0 ≤ b < n, it follows that x an+b λ a x b 1. Hence, k |a and b 0. So, kn |s(q m λ s −1) 0. Conversely, if s(q m λ s −1) 0 mod kn, then x s(q m λ s −1) x ckn λ ck 1.
(ii) This statement follows immediately from Definition 13. (iii) From the given condition it follows that x sq jm β j x s , for 0 ≤ j ≤ l − 1. If β 1, it follows from Definition 13 that p(x) c λ s (x) and that m s m. If β 1 the resulting coefficient of x s is equal to 1 + β + β 2 + · · · + β l−1 1 − β l /1 − β 0. Thus p(x) is the zeropolynomial. (iv) If we define a subpolynomial of a polynomial p(x) as a polynomial not equal to the zero polynomial or to p(x) itself and such that all its terms are also terms of p(x), then the statement is an immediate consequence of Definition 13. It follows that either s 0 or s n/2 and so m s 1, or sq j n − s mod n for some minimal integer j > 0. Hence, s(q j + 1) 0 mod n, and m s 2 j. The only-if-part of the statement is obvious.
We remark that as a consequence of Theorem 15 (v) the number of constacyclonomials is at most equal to the number of cyclonomials c s (x), for fixed values of c λ s (x)n, q and λ. It appears that for λ 1 the first number is the smaller one in many cases.

A bilinear form in R n,q,
In this subsection we present a number of properties of constacyclonomials which they share with cyclonomials. To this end we introduce a bilinear form (,) λ in R n,q,λ , while a polynomial p(x) occasionally will be denoted by p in this context.

Definition 16
For every pair of elements p(x) and q(x) of R n,q,λ a bilinear form ( p, q) λ : is defined, where α is a zero of x n − λ of order kn and ζ a primitive nth root of unity.
One can easily verify that this definition really yields a bilinear form in R n,q,λ with values which do not depend on the choice of α and ζ . In the next theorem the irreducible polynomials P λ t (x) introduced in Eq. (1) will play a role. We know that the degree of P λ t (x), t ∈ T n,q,λ , is equal to m λ t being the size of the constacyclotomic coset C n,q,λ t . The coefficient of its one but highest power x m λ t −1 is denoted by p n,q,λ t or shortly by p λ t (remember the conventions mentioned in Sect. 1). Furthermore, we remind the reader of the fact that the size of the constacyclonomial c λ s (x), s ∈ S n,q,λ , and also of the cyclotomic coset C n,q s is equal to m s . In the next theorem and its proof we shall show that the set A n,q,λ of polynomials (10) is an algebra and that the constacyclonomials c λ s (x) constitute an orthogonal basis of A n,q,λ for fixed values of n, q and λ.
Theorem 17 (Orthogonal basis of constacyclonomials) Let α be a zero of x n − λ of order kn, where k is the order of λ in G F(q), and let ζ : α k .

(i) The set A n,q,λ is an algebra over G F(q) with basis c λ
s (x) s ∈ S n,q,λ , and it consists of all polynomials p(x) ∈ R n,q,λ which satisfy the relation p(x) q p(x). (ii) For any s ∈ S n,q,λ \{0}, and for any j one has With respect to the bilinear form (11), the constacyclonomials c λ s (x), s ∈ S n,q,λ , form an orthogonal basis of A n,q,λ , such that for any pair j, k ∈ S n,q,λ one has Proof (i) By definition A n,q,λ is spanned by the constacyclonomials c λ s (x), s ∈ S n,q,λ . All polynomials p(x)∈ A n,q,λ satisfy p(x) q p(x) and A n,q,λ is a vector space. To see this in detail one should apply the property (βp 1 (x) + γ p 2 (x)) q βp 1 (x) q + γ p 2 (x) q for all β, γ ∈ G F(q). An exhaustive construction of constacyclonomials by applying Definition 13 for fixed values of n, q and λ, shows that together these polynomials contain any power x i , 0 ≤ i ≤ n − 1, at most once. So, they are independent and they constitute a basis of A n,q,λ . On the other hand, let p(x) ∈ R n,q,λ be a polynomial such that p(x) q p(x), and let βx s be one of its terms. It follows from the exhaustive construction that there is precisely one constacyclonomial which contains the x-power x s . By multiplying with an appropriate factor and adjusting its label, we may denote this polynomial by c λ s (x). Now, p 1 (x) : p(x) − βc n,q,λ s (x) also satisfies p 1 (x) q p 1 (x), and so we can continue this process, leading to p 2 (x) : p(x)−βc n,q,λ s (x)−γ c n,q,λ u (x). Proceeding in this way, we finally get the zeropolynomial. We conclude that p(x) βc n,q,λ s (x) + γ c n,q,λ u (x) + · · · ∈ A n,q,λ . That A n,q,λ is closed under multiplication is a consequence of the relation ( p 1 (x) p 2 (x)) q p 1 (x) p 2 (x). (ii) If s 0, the polynomial c λ s (x) is a sum of terms α l x l , α l ∈ G F(q) * , where l runs through a subset U of { 1, 2, . . . , n − 1}. So, any term α l x l occurring in c λ s (x) contributes to the sum n−1 i 0 c λ s (α j ζ i ) an amount of α l α jl (1 + ζ l + · · · + ζ (n−1)l ), which is equal to zero for all l ∈ U . If s 0, one obtains n−1 i 0 c λ 0 (α j ζ i ) 1 + 1 + · · · + 1 n.
Here we used m n− j m j . So, the coefficient of x 0 in the rhs is equal to λ+λ q +· · ·+λ q m j −1 m j λ. Hence, the result in this case is (c λ * j , c λ j ) λ nm j λ, since n−1 i 0 c λ 0 (αζ i ) n because of part (ii) of this theorem. Next, we assume that c λ j (x) is self conjugated, and so (c λ * j , c λ k ) λ (c λ j , c λ k ) λ . Like before we may conclude that for k j the rhs is equal to 0, since the inverses of the x-powers in c λ j (x) are in c λ j (x) itself and not in c λ k (x). Let k j. Since c λ j (x) is assumed to be self conjugated, it contains pairs of terms x jq i and x (n− j)q i . Hence, m j is even and j(q m j /2 + 1) 0 mod n, or m j 1 and j ∈ {0, n/2}. Consequently, if m j is even, the polynomial c λ j (x) contains the term λ a−1 x n− j λ a x − j with a j : j(q m j /2 +1)/n. So, in the product (x j +x jq +· · ·+λ a x − j +λ aq x − jq +· · ·)(x j +x jq +· · ·+λ a x − j +λ aq x − jq + · · ·), the coefficient of x 0 is equal to λ a + λ aq + · · · + λ aq m j −1 m j λ a , and the result follows in the same way as before. The case j 0, m 0 1 is covered by the general result with a 0 0, while for j n/2, m n/2 =1 we have a n/2 n 2 (q [1/2] + 1)/n 1 which yields also the correct answer.

An orthogonal transformation matrix
In this section we shall show that the primitive idempotents θ t (x), t ∈ T n,q,λ , form an alternative orthogonal basis for A n,q,λ . It then follows that the transformation matrix between this basis and the orthogonal basis of constacyclonomials is an orthogonal matrix.

An orthogonal basis of primitive idempotent polynomials
Theorem 18 (Orthogonal basis of primitive idempotents) (i) With respect to the bilinear form (11) the primitive idempotents θ t (x), t ∈ T n,q,λ , form an orthogonal G F(q)-basis of the vector space A n,q,λ , satisfying (θ t , θ u ) λ m λ t δ t,u for all t ∈ T n,q,λ . (ii) The number n 0 of irreducible polynomials P λ t (x) and the number of primitive idempotents θ t (x), t ∈ T n,q,λ , are both equal to the number of constacyclonomials c λ s (x), s ∈ S n,q,λ .
Proof (i) From their definition we know θ t (x) 2 θ (x), and hence θ t (x) q θ t (x) for all t ∈ T n,q,λ . So, all θ t (x) belong to A n,q,λ by Theorem 15 (ii). From Theorem 2 (ii) and Theorem 8 (i) it follows that θ t (αζ i ) is equal to 1 if i ∈ C n,q,λ t and equal to 0 otherwise. Hence, (θ t , θ u ) λ n−1 i 0 θ t (αζ i )θ u (αζ i ) is equal to 0 for t u and equal to m λ t for t u. To show that A n,q,λ is spanned by the idempotent polynomials θ t (x), t ∈ T n,q,λ , we assume that p(x) ∈ A n,q,λ is orthogonal to all θ t (x). So, n−1 i 0 θ t (αζ i ) p(αζ i ) 0 for all t ∈ T n,q,λ . Applying Theorem 2 (ii) then yields i∈C n,q,λ t p(αζ i ) 0, t ∈ T n,q,λ . From Theorem 15 (ii) we have that p(x) q p(x), or equivalently p(x q ) p(x). Since for any pair i, j ∈ C n,q,λ t there is a positive integer a such that αζ i (αζ j ) q a , we can write i∈C n,q,λ t p(αζ i ) m λ t p(αζ j ) for some j ∈ C n,q,λ t and for all t ∈ T n,q,λ . It follows that p(αζ i ) 0 for 0 ≤ i ≤ n − 1, and since the degree of p(x) is less than n, we conclude that p(x) 0. Hence, the polynomials θ t (x) form an orthogonal basis of A n,q,λ . (ii) Since the basis of primitive idempotents θ t (x), t ∈ T n,q,λ , and the basis of constacyclonomials c λ s (x), s ∈ S n,q,λ , must have the same number of elements, it follows that n 0 : T n,q,λ S n,q,λ (cf. also Theorem 5 (i)). Furthermore, there is a one-one correspondence between the primitive idempotent θ t (x) and the irreducible polynomial P λ t (x) with zeros αζ t for t ∈ T n,q,λ .
Since the constacyclonomials constitute an orthogonal basis for A n,q,λ , each element p ∈ A n,q,λ can be developed as (cf. Theorem 17 (iii)) In particular we can write for the primitive idempotent θ t (x), t ∈ T λ , the expression Theorem 19 (Orthogonality relations for primitive idempotents) (i) Let μ s,t stand for the sum of the s-powers of the zeros of P λ t (x), for s ∈ S n,q,λ and t ∈ T n,q,λ . Then the coefficients of the idempotent θ t (x) can be written as ξ for t, u ∈ T n,q,λ and s, r ∈ S n,q,λ .
Proof (i) We know that θ λ t (αζ i ) is equal to 1 if αζ i is a zero of the irreducible polynomial P λ t (x), while it is equal to 0 otherwise. Let c λ s (x) be self conjugated. Then c λ * s (αζ i ) c λ s (αζ i )=(αζ i ) s + (αζ i ) sq + (αζ i ) sq 2 · · ·, and hence (c λ * s , θ λ t ) λ where the summation indices i in the m s terms of the rhs run through the set C n,q,λ t . Since αζ i is a zero of P λ t (x), we have that (αζ i ) q j is also a zero of P λ t (x) and we can write The mapping i → i is one-to-one, and so the m s summations are all equal to i∈C n,q,λ t (αζ i ) s . The individual terms in this summation are the s -powers of the zeros of P λ t (x), and they are zeros themselves of some irreducible polynomial P λ s t s (x). So, ξ t s m s μ s,t /nm s λ a s = μ s,t /nλ a s . If c λ s (x) is not self conjugated, we have c λ * s (x) c λ n−s (x) . The result then follows in a similar way and by using a n−s a s 1 in this case. We remind the reader that all integers which occur in Theorem 19 are to be taken in G F(q). In case that a denominator m i or m λ t is equal to zero modulo q, one must consider this variable in connection with the numerator of the fraction to which it belongs to get an equality which makes sense. E.g. the fraction w s ξ t s m λ t m s μ s,t nm λ t in (iii) appears to be a well defined integer by applying (ii). Another remark is that for q 2 and odd n, Theorem 19 (i) delivers the well-known result for primitive idempotents in the binary case [18, Ch. 8, Theorem 6].

Idempotent tables Ξ n,q,λ and M n,q,λ
We shall reformulate now the orthogonality relations of Theorem 19 (iii) in terms of matrices.

Definition 20 (Definition of primitive idempotent table for constacyclic codes)
The n 0 × n 0matrix Ξ n,q,λ over G F(q) has elements Ξ n,q,λ s,t : ξ t s , s ∈ S n,q,λ , t ∈ T n,q,λ . The adjoint matrix Ξ n,q,λ * is the matrix with elements Ξ n,q,λ * s,t
The next simple example will enable the reader to verify all properties stated in Theorems 18 and 19.
We introduced Ξ n,q,λ as the transformation matrix from one orthogonal basis to another. However, because of its orthogonality over G F(q), we could equally well consider Ξ n,q,λ , for any relevant triple (n, q, λ), as a primitive idempotent table, (shortly idempotent table) resembling the irreducible character tables for finite groups (cf. e.g. [7,13]). In this picture the columns of the table Ξ n,q,λ , which represent the primitive idempotents with labels t ∈ T n,q,λ , being the indices of the constacyclotomic cosets C n,q,λ t , correspond to irreducible characters. The labels s ∈ S n,q,λ of the rows are the indices of those cyclotomic cosets C n,q s which afford a constacyclonomial. These constacyclonomials or these cosets can be seen as the counterparts of the classes of conjugated elements in a finite group. Instead of Ξ n,q,λ , we shall mostly consider the matrix M n,q,λ (e.g. in Example 22) with elements μ s,t nλ a s Ξ n,q,λ where s * s when c λ s (x) is self conjugated, and s * n − s otherwise (cf. Theorem 19 (i)).

The case of cyclic codes
The analogy between the two types of tables, mentioned in the previous section, is even stronger for λ 1, i.e. in the case of cyclic codes. In this section we shall take a closer look at this case.

Primitive idempotent tables M n,q
For the sake of simplicity, we choose the ζ -representation for the zeros of x n − 1 and omit the parameter value λ 1 (cf. Remark 9). So, instead of the sets S n,q,1 and T n,q,1 we take the index sets S n,q and T n,q . These two sets can be chosen identical. In order to establish more similarities with character tables, we defined in Sect. 4 C n,q t * : C n,q −t as the conjugated cyclotomic coset of C n,q t , t ∈ T n,q , where t * : n − t is an integer in [0, n − 1]. It will be obvious that m t * m t . Correspondingly, we define P t * (x) of degree m t as the conjugated irreducible polynomial of P t (x) and θ t * (x) as the conjugated primitive idempotent of θ t (x). Actually, the polynomial P t * (x) is the monic reciprocal of P t (x), formally expressed by which was introduced in Theorem 2 (vi) for any P λ t (x), t ∈ T n,q,λ . We say that . We also introduced in Definition 14 the constacyclonomial c λ * s (x) as the conjugate of c λ s (x) . We now write this polynomial as c λ s * (x), so s * s if C λ s (x) is self conjugated and s * n − s otherwise. Because of these decisions and definitions, and because the s-powers of the zeros of any irreducible polynomial contained in x n − 1 are the zeros of some other (or the same) irreducible factor of x n − 1, we can simplify and extend the relations of Theorem 19 as follows.

Theorem 23
(i) The coefficients of the idempotent θ t (x) for a cyclic code can be written as ξ t s μ s,t /n, where μ s,t stands for the sum of the s-powers of the zeros of P t (x), for all s ∈ S n,q and t ∈ T n,q . Theorem 24 (Primitive idempotent table for cyclic codes) The entriesμ s,t of the table M n,q , s ∈ S n,q , t ∈ T n,q ( S n,q ), satisfy the following properties.
(i) s∈S n,q m s m t μ s,t μ s * ,u nδ t,u , t∈T n,q m s m t μ s,t μ r * ,t nδ s,r , (ii) μ s * ,t μ s,t * , m s μ s,t m t μ t,s . (iii) μ s,0 1 for all s ∈ S n,q and μ 0,t m t for all t ∈ T n,q . (iv) If n is even μ s,n/2 (−1) s for all s ∈ S n,q and μ n/2,t (−1) t m t for all t ∈ T n,q .
Proof (i) and (ii) These relations follow immediately from the orthogonality relations in Theorem 19 (iii) and from the equality for μ s,t in Theorem 19 (ii).
(iii) and (iv) follow from Theorem 19 (ii) by substituting respectively p 0 −1 and p n/2 1. These values are yielded by the irreducible polynomials P 0 (x) x − 1 and P n/2 (x) x + 1.
The statements in Theorem 23 provide us with a link to the theory of idempotents of cyclic codes as developed in [25]. The column of M n,q with index 0 (its 'first' column) is the all-one column, and so θ 0 corresponds to the trivial character χ 1 of a finite group G. Furthermore, the row of M n,q with index 0 (its 'first' row) contains all values m t , i.e. the sizes of the cyclonomials. One could consider these values as counterparts of the dimensions (degrees) χ j 1 of the irreducible representations of a finite group . Now, the second orthogonality relation of Theorem 23 (i) gives for s r 0 that t∈T n,q,1 m t n. It is tentative to see this elementary equality as the counterpart of the well known Burnside relation j (χ j 1 ) 2 n, which results from a similar orthogonality relation for character tables. All these similarities with irreducible character tables, strengthen the introduction of the name of primitive idempotent table for the matrix M n,q , and more in general for M n,q,λ (cf. Sect. 5). We remark that the similarities between the orthogonality relations and their consequences for idempotent generators on the one hand and irreducible characters on the other, will not come as a surprise if one realizes that both topics can be embedded in the general theory of idempotents for semi-simple algebras (cf. [6,23,24]).

Blocks of conjugated cyclonomials and idempotents
Inspired by the previous remarks we introduce the following notions. Let r be an element of the multiplicative group U n consisting of the positive integers modulo n which are prime to n. It will be obvious that the set rC n,q s is identical to the cyclotomic coset C n,q rs . We shall call it the r -conjugate of C n,q s . Similarly, the cyclonomial c rs (x) is the r -conjugate of c s (x), and the irreducible polynomial P rt (x) the r -conjugate of P t (x). Since (n, r ) 1, one easily proves that m rs m s , that c rs (x) and c s (x) have the same size, and that P rt (x) and P t (x) have the same degree. For r n − 1 ( −1 mod n) we obtain the notions of conjugated irreducible polynomial and conjugated cyclonomial which were introduced already earlier in this text, and which correspond to the notion of conjugated cyclotomic coset at the end of Sect. 4. We say that c s (x) is r -self conjugated if c rs (x) c s (x), and that P t (x) is r -self conjugated if P rt (x) P t (x). If θ rt (x) is the primitive idempotent generated by P rt (x), then θ rt (x) and θ t (x) are also said to be r-conjugated, and if they are equal θ t (x) is said to be r-self conjugated. There exists a simple relationship between the primitive idempotent θ t (x) and its r-conjugate θ rt (x). Let 1/r be the inverse of r in U n . From Theorem 2 (ii) it follows that θ t (x 1/r ) 1 for x β r and β is a zero of P t (x), and that θ t (x 1/r ) 0 for x β r and β is a zero of P u (x), u t. We conclude that for all t ∈ T n,q θ rt (x) θ t (x 1/r ), r ∈ U n .
Furthermore, the set of all cyclonomials is denoted by Cy n,q : {c s (x) |s ∈ S n,q }, (17) and the set of all primitive idempotent generators by I d n,q : {θ t (x) |t ∈ T n,q }.
Let H be the subgroup of U n generated by q. Remember that (n, q) 1, so q i ∈ U n for all i. It will be clear that the elements of H are the same as those of the cyclotomic coset C n,q 1 , and so |H | m 1 . Since |U n | ϕ(n), the quotient group U n /H has a : ϕ(n)/m 1 elements and we write U n H r 1 ∪ H r 2 ∪ · · · ∪ H r a , with H r i : r i H , for 1 ≤ i ≤ a, and H r 1 : H . The elements r i are determined up to powers of q. Since the same holds for the integers in S n,q , we can chose r 1 ( 1), r 2 , . . . , r d such that they are the integers of the set S n,q 1 S n,q ∩ U n . More generally, we define for each divisor d ≤ n of n S n,q d : s s ∈ S n,q , (n, s) d .
We remark that for any s ∈ S n,q d , any integer i ∈ C n,q s satisfies (n, i) d, since (n, q) 1 . Hence, the cyclotomic cosets defined by (20) together contain all integers i ∈ [0, n − 1] with (n, i) d, and so S n,q d S n,q d .
We also remind the reader that the elements of any cyclotomic coset C n,q i can be obtained from the subgroup C The expression in the rhs of (22) stands for the multiset where each integer of C n,q i occurs m 1 /m i times (cf. also Theorem 8 (v)). The above lines show that U n induces a group G of permutations on the set of cyclotomic cosets by means of the transformation C n,q s → C n,q rs , and hence, that U n also induces permutation groups G and G acting on the sets Cy n,q and I d n,q . These groups are isomorphic and the orbits of G and G are called blocks of cyclonomials and blocks of idempotents. The subgroup H of U n contains precisely those elements which induce the identity permutation on these sets. Because T n,q is identical to S n,q , we define T n,q d : S n,q d for all d| n. For even n, there exists another permutation group acting on the set of cyclotomic cosets, and hence on Cy n,q and I d n,q , induced by i → i + n/2 mod n for all integers i ∈ {0, 1, . . . , n − 1}. One can easily verify that this operation transforms C n,q s into C n,q s+n/2 . The orbits of this group have size 2 or 1. In the first case we shall speak of pairs of associated cyclotomic cosets, associated cyclonomials and associated idempotents. In the second case all these objects are called self associated.

Theorem 25
The primitive idempotents for cyclic codes C n,q , (n, q) 1, have the following properties.

The case of negacyclic codes
In this final section we focus on the class of negacyclic codes, so we take λ −1 and k 2.

Primitive idempotent tables M n,q,−1
the next we shall restrict the first transformation to those cyclotomic cosets which correspond