Primitive idempotent tables of cyclic and constacyclic codes

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(1)Delft University of Technology. Primitive idempotent tables of cyclic and constacyclic codes van Zanten, A. J. DOI 10.1007/s10623-018-0495-0 Publication date 2018 Document Version Final published version Published in Designs, Codes, and Cryptography. Citation (APA) van Zanten, A. J. (2018). Primitive idempotent tables of cyclic and constacyclic codes. Designs, Codes, and Cryptography, 87 (2019), 1199–1225. https://doi.org/10.1007/s10623-018-0495-0 Important note To cite this publication, please use the final published version (if applicable). Please check the document version above.. Copyright Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.. This work is downloaded from Delft University of Technology. For technical reasons the number of authors shown on this cover page is limited to a maximum of 10..

(2) Designs, Codes and Cryptography (2019) 87:1199–1225 https://doi.org/10.1007/s10623-018-0495-0. Primitive idempotent tables of cyclic and constacyclic codes A. J. van Zanten1,2 Received: 24 January 2018 / Revised: 5 April 2018 / Accepted: 15 May 2018 / Published online: 9 July 2018 © The Author(s) 2018. Abstract For any λ ∈ G F(q)∗ a λ-constacyclic code C n,q,λ : g(x), of length n is a set of polynomials in the ring G F(q)[x]/x n − λ, which is generated by some polynomial divisor g(x) of x n −λ. In this paper a general expression is presented for the uniquely determined idempotent n,q,λ n,q,λ (x), where Pt (x) is generator of such a code. In particular, if g(x) : (x n − λ)/Pt n an irreducible factor polynomial of x − λ, 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) with (n, q) 1 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 (cyclic codes) and λ −1 (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). Keywords Constacyclic codes · Cyclic codes · Negacyclic codes · Idempotent generating polynomials · Semisimple rings · Irreducible character tables Mathematics Subject Classification 12E05 · 12E20 · 16S34 · 20C05 · 20C15 · 20C20 · 94B15 · 94B60. 1 Introduction A λ-constacyclic (shortly constacyclic) code C n,q,λ : g(x) mod x n − λ of length n is generated by some polynomial divisor g(x) over G F(q) of x n − λ, with λ ∈ G F(q)∗ . So, C n,q,λ is a set of polynomials in the ring R n,q,λ : G F(q)[x]/x n − λ. For λ 1 one obtains the family of cyclic codes which are well known [18, 19, 24]. For λ −1 one obtains the. Communicated by I. Landjev.. B. A. J. van Zanten a.j.vanzanten@ewi.tudelft.nl. 1. Faculty of Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands. 2. Department of Communication and Informatics, University of Tilburg, Tilburg, The Netherlands. 123.

(3) 1200. A. J. van Zanten. 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 n,q,λ xn − λ Pt (x), (1) t∈T n,q,λ. where T n,q,λ is an index set containing the indices of all these irreducible polynomials. If n,q,λ (x) for some fixed t ∈ T n,q,λ , then the code f t (x) is called a f t (x) : (x n − λ)/Pt minimal or irreducible constacyclic code. In algebraic terms, such a code is a minimal ideal in n,q,λ (x) is called a maximal constacyclic code. An idempotent the ring R n,q,λ . The code Pt polynomial in R n,q,λ is a polynomial en,q,λ (x) ∈ R n,q,λ with the property that en,q,λ (x)2 en,q,λ (x).. (2). It will be clear that if (2) holds, then all positive powers of en,q,λ (x) are identical. If en,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 θt (x) and those of the maximal constacyclic codes by ϑt (x), t ∈ T n,q,λ [16, 22, 25]. The polynomials θt (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)∗ .. (3). 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 An,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 An,q,λ . The orthogonal transformation matrix between these two bases can be interpreted as an orthogonal table of. 123.

(4) Primitive idempotent tables. 1201. 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 n,q,λ of convenience. So, we shall write eλ (x) instead of en,q,λ (x), Ptλ (x) for Pt (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 n,q,λ R n,q,λ , An,q,λ , S n,q,λ , T n,q,λ , C n,q,λ and Ct .. 2 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 eλ (x) is the uniquely determined idempotent generator of C , then there exist polynomials p(x) and q(x) such that eλ (x) p(x)g(x) and g(x) q(x)eλ (x) in R n,q,λ . (ii) eλ (αζ i ) 0 if g(αζ i ) 0 and eλ (αζ i ) 1 if h(αζ i ) 0 for i ∈ {0, 1, . . . , n − 1}. (iii) If eλ (x)∗ is the idempotent generator of the code C ∗ : h(x) , then eλ (x)+eλ (x)∗ 1. (iv) c(x) ∈ C if and only if eλ (x)c(x) c(x). (v) If C1 and C2 are λ -constacyclic codes with idempotent generators e1λ (x) and e2λ (x) , then C1 ∩ C2 and C1 + C2 are also λ -constacyclic codes with idempotent generators e1λ (x)e2λ (x) and e1λ (x) + e2λ (x) − e1λ (x)e2λ (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 {θt (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: (i) θt (x)θu (x) 0 if t u and θt (x)2 θt (x); (ii) θt (αζ i ) 1 if αζ i is a zero of Ptλ (x) , while θt (αζ i ) 0 if αζ i is a zero of Puλ (x), u t;. 123.

(5) 1202. A. J. van Zanten. (iii) θ i1 (x) + θi2 (x) + · · · + θir (x) is the idempotent generator of the constacyclic code f i1 (x) f i2 (x) . . . f ir (x) ; (iv) ϑt (x) 1 − θt (x) and t∈T n,q,λ θt (x) 1; n,q,λ , then there (v) If eλ (x) is the idempotent generator of some λ -constacyclic code in R exist elements ξ1 , ξ2 ,…., ξr ∈ G F(q) such that eλ (x) ri1 ξi θi (x). (vi) Let Ptλ (x), t ∈ T n,q,λ , be a monic irreducible polynomial of degree m λt , then its −1 ∗ λ reciprocal Ptλ (x) : Ptλ (0)−1 x m t Ptλ (1/x) is a monic irreducible polynomial Ptλ∗ (x), −1 −1 t ∗ ∈ T n,q,λ , such that the zeros of Ptλ∗ (x) are the inverses of the zeros of Ptλ (x) . The corresponding primitive idempotent generator satisfies θt ∗ (x) : λx n θt (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 Ptλ∗ (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)Ptλ (x) for some polynomial q(x) ∈ λ λ ∗ R n,q,λ . From this equality we derive 1 − λx n x n−m t q(1/x)x m t Ptλ (1/x) q0 (x)Ptλ (x), −1 ∗ ∗ with qo (x) ∈ R n,q,λ . Hence, x n − λ−1 −λ−1 qo (x)Ptλ (x) and Ptλ∗ (x) :Ptλ (x) is a n −1 n −1 −1 divisor of x − λ . Finally, x θt (1/x)has value λ · 1 λ when we substitute for x −1 a zero of Ptλ∗ (x), and it has value λ · 0 0 when substituting one of the other zeros of n −1 x − λ . So, λx n θt (1/x) must be identical to θt ∗ (x). 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. Theorem 3 (i) If g(x) 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 eλ (x) (nλ)−1 xh (x)g(x) in R n,q,λ , where h (x) is the formal derivative of the checkpolynomial h(x) : (x n − λ)/g(x). (ii) The idempotent generator of the dual code g(x)∗ is given by eλ (x)∗ (nλ)−1 xg (x)h(x). 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) and b(x) such that a(x)g(x) + b(x)h(x) 1. Multiplying by a(x)g(x) yields (a(x)g(x))2 + a(x)b(x)g(x)h(x) a(x)g(x), and therefore (a(x)g(x))2 a(x)g(x) mod x n −λ. So, we can write eλ (x) a(x)g(x). To determine a(x), we take derivatives of both sides of the relation g(x)h(x) x n − λ, yielding h (x)g(x) + h(x)g (x) nx n−1 . In R n,q,λ this is equivalent to (nλ)−1 x(h (x)g(x) + h(x)g (x)) 1. 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 eλ (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 t ∈ T n,q,λ , belonging λpolynomials θt (x), λ to the polynomial in (1). So, g(x) Pi (x), h(x) Pt (x), and hence θt (x) it Piλ (x), according to Theorem 3 (i). Applying the rule for determining (nλ)−1 x Ptλ (x) it. 123.

(6) Primitive idempotent tables. the derivative of a product of functions yields. 1203. . t∈T n,q,λ θt (x). . (nλ)−1 x(x n − λ) (nλ)−1 xnx n−1 1 mod x n − λ.. (nλ)−1. t∈T n,q,λ. Ptλ (x). . 3 Constacyclic codes C n,q, for various values of In this section we study the relationship between the λ-constacyclic codes for different values of λ. To this end we shall need the notion of the order of a polynomial p(x) ∈ G F(q)[x], i.e. the least positive integer e such that p(x) is a divisor of x e − 1. A well known property is that the order of a product of polynomials which are pairwise relatively prime, is equal to the least common multiple (lcm) of the orders of its zeros (cf. [17, Theorem 3.9]). Another well known property is that the order of an irreducible polynomial f (x) ∈ G F(q)[x], with f (0) 0, of degree m is equal to the order of any of its zeros in the splitting field G F(q m ) of f (x) over G F(q) (cf. [17, Theorem 3.3]). Theorem 5 Let n be a positive integer and q a prime power with (n, q) 1 . Let F be the smallest extension field of G F(q) such that it contains all zeros of x n −λ , while λ ∈ G F(q)∗ has order k . Let furthermore e be the order of x n − λ in G F(q)[x].. . (i) If the n 0 : T n,q,λ irreducible factor polynomials Ptλ (x) of x n −λ in G F(q)[x] have order et , 1 ≤ t ≤ n 0 , then e is equal to the least common multiple e1 , e2 , . . . , en 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 k−1 n xe − 1 (x − λ j ). j0. (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−b0 all satisfying β (b−b0 )n 1, for some fixed integer b0 , and so these elements form a subgroup of order n. Hence, n is a divisor of e. Furthermore, e/n λe/n β bn β be 1, and hence e akn for some positive integer a. However, kn k since α λ 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. 123.

(7) 1204. A. J. van Zanten. (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 λs 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 λs 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. (iv) Since α d generates a subgroup of order h in G and α n a subgroup of order k, we have k(h, n/d) h. So, kn hd if and only if (h, n/d) n/d, or equivalently, if n/d is a divisor of h. The other results follow easily from (iii). 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 eλ (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 eλ (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 (x n − λ)h(x) (x e − 1)h λ (x). Taking derivatives on both sides of this equality, applying Theorem 3 (i) and dividing by x n − λ, yields the relation kne(x) + nx n t(x) nλt(x)eλ (x) + knx e mod x e − 1, with the polynomial t(x) : k−1

(8) x n i . Since x n − λ divides x e − 1, the above equality also x e − 1/x n − λ λ−1 i0 λ 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 eλ (x) e(x). . 4 Generalization of cyclotomic cosets and cyclonomials From standard results on cyclotomic cosets (cf. [17]) it is well known that the zeros of any m t −1 irreducible factor of x n − 1 can be written as ζ t , ζ tq , . . . , ζ tq 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 ),. 123.

(9) Primitive idempotent tables. 1205. we establish a one-one correspondence between the irreducible polynomials Pt1 (x) and the q-cyclotomic cosets mod n n,q. Ct. (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. n,q,. 4.1 Constacyclotomic cosets Ct. We introduce the integer l : (q − 1)/k, and we define ordered subsets of {0, 1, . . . , n − 1} as follows. Definition 7 For any triple of parameters n, q and λ satisfying condition (3), the set.

(10) n,q,λ : c0 ( t), c1 , . . . , cm λ −1 , Ct t. ci+1 ci q + l. mod n, 0 ≤ i < m λt − 1,. (5) (6). where m λt is the smallest positive integer satisfying cm λ −1 q + l c0 , is called a (q-) constat cyclotomic 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

(11) properties we shall use the notation. n,q,λ n,q,λ aCt +b : aCt +(b, b, . . . , b), which stands for ac0 + b, ac1 + b, . . . , acm λ −1 + b , t where all integers aci + b must be computed mod n. 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 Ptλ (x) over G F(q) contained in x n − λ can n,q,λ be written as αζ c where c runs through the set Ct , while m λt is equal to the degree of that polynomial. (ii) The integers ci in (5) satisfy ci tq i + (q i − 1)/k mod n, 0 ≤ i ≤ m λt − 1. n,q,λ is equal to the smallest positive integer which satisfies the relation (iii) The size m λt of Ct λ m t (kt + 1)(q − 1) 0 mod kn. (iv) For any integer b ≥ 0 and for all i, 0 ≤ i ≤ n − 1, one has kci+b + 1 q b (kci + 1) mod kn. n,q,λ n,q,λ (v) Modulo n one has C0 k −1 (0, q − 1, q 2 − 1, . . .), C0 + t(1, q, q 2 , . . .) n,q,λ n,q,λ n,q and kCt + 1 m λt /m kt+1 × Ckt+1 , where the notation a × C n,q,λ m λ0 /m λt × Ct n,q or a × C means that each integer of the relevant coset occurs a times in the multiset at the left hand side of the equality. Proof λ n (i) Let α be a zero of the irreducible factor x − λ of degree m 0 . Then we can

(12) P (x)m of 0 −1 λ q q write P (x) (x − α)(x − α ) . . . x − α . From Theorem 5 we know that i. for all relevant i, we can write α q αζ ci for some integer ci ∈ {0, 1, . . . , n − 1}. i+1 Hence, α q α q ζ qci αα q−1 ζ qci αζ l+qci by using q − 1 kl, and we obtain. 123.

(13) 1206. A. J. van Zanten m. ci+1 l + qci for 0 ≤ i ≤ m 0 − 2. Furthermore, since α q 0 α, we also have l + qcm 0 −1 c0 . So, the relations (4) and (5) with t t0 0 define Ptλ0 (x) : P λ (x).. Next, let t1 be the least integer in {0, 1, . . . , n − 1}\C0 and define Ptλ1 (x) of degree n t m t1 as the irreducible factor of x − λ which has αζ 1 as zero. Similarly as before it appears that this polynomial is defined by (4) and (5) with t t1 . Proceeding in this way until all integers of {0, 1, . . . , n − 1} have been dealt with, we end up with an index set T n,q,λ {t0 ( 0), t1 , . . .} ⊂ {0, 1, . . . , n − 1}, such that any irreducible polynomial contained in x n − λ can be indexed by some integer in T n,q,λ and vice versa. (ii) This relation can easily be proved by incomplete induction on i, 0 ≤ i ≤

(14) m λt − 1.

(15) n,q,λ. λ. λ. λ. λ. (iii) From (ii) we have that cm λ l + tq m t + l q m t −1 + · · · + q tq m t + q m t − 1 /k. t By requiring cm λ t, we obtain the relation in (iii). t (iv) By iteration we get ci+b l + lq + · · · + lq b−1 + q b ci (q b − 1)/k + q b ci mod n. Hence, kci+b + 1 q b + kq b ci mod kn. (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 n,q,1 not provide us with the cyclotomic cosets (4). In terms of the integers of Ct (c0 ( t), c1 , . . . , cm t −1 ), the zeros of the corresponding irreducible polynomial can be written as αζ c0 , αζ ci , …, αζ cm 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 , m t −1 ζ tq , . . . ζ tq , the ζ -representation. Application of Theorem 8 (v) with λ 1 and k 1, shows that the two types of cosets are related by n,q,1. n,q. Ct−1 + 1 Ct .. (7) n,q,1. , This relation implies that m 1t−1 m t for all t ∈ T n,q . In the next we keep calling Ct n,q t ∈ T n,q,1 , a constacyclotomic coset and Ct , t ∈ T n,q , a cyclotomic coset. Furthermore, as n,q,λ + 1 is, strictly speaking, an ordered was already remarked in Theorem 8 (v), the set kCt 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. 4C014,5,2 + 1 (1, 5, 11, 13, 9, 3) and 4C414,5,2 + 1 (3, 1, 5, 11, 13, 9). n,q,μ. Next, we present a theorem which shows how to determine constacyclotomic cosets Ct 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 Ptλ (x) (cf. Theorem 8), for t ∈ T n,q,λ and for s ≥ 0. This s polynomial is an irreducible factor of x n − λs , denoted by Ptλs (x). Theorem 10 Under the conditions of Theorem 8, the following relations hold. (i) In G F(q)[x] one has the factorization x kn − 1 . k s1. (x n − λs ).. (ii) The zeros of x n − λs , 1 ≤ s ≤ k, can be written as α ki+s or, equivalently, as α s ζ i , 0 ≤ i ≤ n − 1. n,q,λs (iii) The integers of the constacyclotomic coset Ct (c0 ( t), c1 , . . . , ctm−1 ), m : s λ m t , satisfy the recurrence relation kci+1 + s (kci + s)q mod kn, 0 ≤ i ≤ m − 1.. 123.

(16) Primitive idempotent tables. 1207. (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 s n,q,λs kn,q Ct , t ∈ T n,q,λ , and the cyclotomic cosets Ckt+s , kt + s ∈ T kn,q . For s 0 n,q the mapping yields a one-one correspondence between the cyclotomic cosets Ct , kn,q n,q kn,q . t ∈ T , and the cyclotomic cosets Ckt , kt ∈ T n,q,λ i , of the irreducible polynomial (v) For any s ≥ 0 the s-powers of the zeros αζ , i ∈ Ct s Ptλ (x), t ∈ T n,q,λ , are zeros of the irreducible polynomial Ptλs (x), where ts (k, s)t is an index in the α , ζ -representation with α α s and ζ ζ s/(k,s) . Each zero of s s Ptλs (x) corresponds to m λt /m λts zeros of Ptλ (x). Proof (i) 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 λs is a divisor of k, and so the order of x n − λs 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 λs . s (iii) Let α s ζ c0 , α s ζ c1 , … , α s ζ cm−1 be the zeros of the irreducible polynomial Ptλ (x) of i i+1 degree m. Then we can write (α s ζ c0 )q α s ζ ci , 0 ≤ i ≤ m − 1. So, (α s ζ c0 )q i+1 (α s )q ζ qci α sq α kqci α (kci +s)q . On the other hand, (α s ζ c0 )q α s ζ ci+1 α kci+1 +s , and so kci+1 + s (kci + s)q mod kn. kn,q (iv) Let Ca 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 n,q,λs kn,q so there is precisely one constacyclotomic coset Ct which is mapped to Ca . If kn kt n 0 s 0 the zeros of x − 1 are written as α , and hence the zeros of x − λ 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 − λs . Let αζ i be a zero of Ptλ (x), s then α s ζ is is a zero of some irreducible polynomial Ptλs (x), and this polynomial is the. . We put α : α s and ζ : α k/(k,s) , where k/(k, s) is the order same for all i ∈ Ct s s of λ in G F(q), and so ζ α sk/(k,s) ζ s/(k,s) . It follows that the zeros of Ptλs (x) can n,q,λ. n,q,λs. n,q,λs. , with ts (k, s)t. Each integer in Cts be written as α ζ j , j ∈ Cts s n,q,λ m λt /m λts times to some integer in Ct .. corresponds . Example 11 Take n 8, q 5 and λ 2. It follows that k : ord5 (2) 4. Since x 8 − 2 does not divide x 16 − 1, its order is kn 32. We have the following factorization x 32 − 1 . k−1 . (x 8 − λs ) (x 8 − 2)(x 8 − 4)(x 8 − 3)(x 8 − 1).. s0. The 5-cyclotomic cosets modulo 32 are C032,5 (0), C132,5 (1, 5, 25, 29, 17, 21, 9, 13), (2, 10, 18, 26), C332,5 (3, 15, 11, 23, 19, 31, 27, 7), C432,5 (4, 20), C832,5 (8), 32,5 32,5 (16), C12 (12, 28) and C24 (24). The factorization of the polynomials x 8 −λs , s 0 and s 2, into irreducible polynomials over G F(5) is respectively x 8 − 1 (x + 1)(x − 1)(x + 2)(x − 2)(x 2 + 2)(x 2 − 2) and x 8 − 4 (x 4 + 2)(x 4 − 2), while x 8 − 2 and x 8 − 3 are irreducible themselves. Only C132,5 contains integers equal to 1( s) modulo 4( k). So, there is only one constacyclotomic coset. C232,5 32,5 C16. 123.

(17) 1208. A. J. van Zanten. in the case s 1. Subtracting 1 from the integers in C132,5 and next dividing the results by 4, provides us with C08,5,2 (0, 1, 6, 7, 4, 5, 2, 3) (cf. Theorem 10 (iv)). Similarly, for s 2, we obtain C08,5,4 (0, 2, 4, 6) and C18,5,4 (1, 7, 5, 3) from C232,5 and C632,5 , respectively. In the case s 3, the cyclotomic coset C332,5 delivers C08,5,3 (0, 3, 2, 5, 4, 7, 6, 1). For s 4 we obtain from the cyclotomic cosets Ci32,5 , i ∈ {0, 4, 8, 12, 16, 24}, the constacyclotomic cosets C78,5,1 (7), C08,5,1 (0, 4), C18,5,1 (1), C28,5,1 (2, 6), C38,5,1 (3) and C58,5,1 (5). Finally, for s 0 we find the cyclotomic cosets C08,5 (0), C18,5 (1, 5), 32,5 C28,5 (2), C38,5 (3, 7), C48,5 (4) and C68,5 (6) from C032,5 , C432,5 , C832,5 , C12 , 32,5 32,5 C16 and C24 . The cosets in the cases s 4 and s 0 are related by (7). To illustrate Theorem 10 (v), we take the irreducible polynomial P08,5,2 (x) x 8 − 2. Let α be one of its zeros of order 32. Then the complete set of zeros can be written as αζ i , i ∈ C08,5,2 , with ζ α k α 4 . The 2-powers of these zeros are α 2 , α 2 ζ 2 , α 2 ζ 4 and α 2 ζ 6 , and each of them 22 occurs twice. These 2-powers are zeros of P(4,2).0 (x) P04 (x) x 4 − 2, since the zeros of that polynomial are determined by C08,5,4 (0, 2, 4, 6). To see this one has to apply relation (6) with l (q − 1)/k 4/2 2. Theorem 12 (i) Let r be a fixed integer with (r , n) 1. Let a satisfy 0 ≤ a < n and n,q,λ n,q,λ ka − r + 1 0 mod n/(l, n). Then Cr t+a rCt + (a, a, . . .) with m r t+a m t , and the mapping t → r t + a mod itself induces a permutation of n of [0, n − 1] onto n,q,λ. order at most (l, n) on the set Ct. t ∈ T n,q,λ . n,q,λ. n,q,λ. (ii) If a n/(l, n), then Ct+a Ct + (a, a, . . .), and the mapping t → t + a mod n n,q,λ defines a permutation on {Ct |t ∈ T n,q,λ } of order at most(l, n). n,q,λ n,q,λ (iii) If a satisfies ka + 2 0 mod n/(l, n), then C−t+a −Ct + (a, a,. . . .), andthe n,q,λ. mapping t → −t + a mod n defines a permutation on the set Ct. t ∈ T n,q,λ of order at most 2. n,q,λ. Proof (i) We know from (6) that the elements of Ct satisfy ci+1 qci + l mod n for n,q,λ λ 0 ≤ i ≤ m t −1. Now, we define di : r ci +a mod n. If we require that the elements of Ct keep their mutual order under the mapping on Crλt+a , we must have that di+1 r ci+1 + a mod n. Consequently, di+1 − qdi − l r ci+1 + a − rqci − 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 n,q,λ by r does not alter the size of the coset, and neither does adding the the integers ci of Ct same integer a to all r ci 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 n,q,λ that the mutual order of the integers in Ct is not to be changed by the mapping. An example is provided by the constacyclotomic cosets C012,7,2 (0, 2, 4, 6, 8, 10), C112,7,2 (1, 9, 5) and C312,7,2 (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. 123.

(18) Primitive idempotent tables. 1209. n,q,1. set Ct. t ∈ T n,q,1 . By applying (7), one can see that this is equivalent to a permutation n,q. on the set Ct 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 n,q,−1. n,q,1 n,q,1 Ct : Cn−t−2 as the conjugated constacyclotomic. t ∈ T n,q,−1 . We define Ct ∗ n,q,1. n,q. n,q. n,q. coset of Ct , and equivalently Ct ∗ : Cn−t as the conjugated cyclotomic coset of Ct . n,q,−1 n,q,−1 n,q,−1 Similarly, Ct ∗ : Cn−t−1 is the conjugated constacyclotomic coset of Ct . n,q,. 4.2 Constacyclonomials cs. (x). A second notion in the theory of cyclic codes that we shall generalize is that of cyclonomic n,q polynomial or cyclonomial (cf. e.g. [25]). To each cyclotomic coset Cs of size m s there corresponds a cyclonomial n,q. cs (x) : x s + x sq + · · · + x sq. m s −1. mod x n − 1.. (8). Clearly, such a polynomial, shortly written as cs (x), (cf. Sect. 1) has the property cs (x)q cs (x). mod x n − 1.. (9). In the following definition is s an integer of {0, 1, . . . , n − 1}, and λ an arbitrary element of G F(q)∗ . m λ −1. Definition 13 The polynomial csλ (x) : x s + x sq + · · · + x sq s mod x n − λ in R n,q,λ is λ λ called a monic constacyclonomial of size m s, if it is not the zero polynomial and if m s is the smallest positive integer such that x sq. mλ s −1. q. x s mod x n − λ.. Since β q β, for any β ∈ G F(q)∗ , we could call any polynomial βcsλ (x) with csλ (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 cs1 (x) ≡ cs (x). It will be obvious that if csλ (x) contains a term βx t , β ∈ G F(q)∗ , then ctλ (x) β −1 csλ (x), and so ctλ (x) and csλ (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 csλ (x) does not contain a term βx n−s . Then it follows easily λ (x) is a different constacyclonomial of the same size. The monic from Definition 13 that cn−s λ λ (x) are called a pair of conjugated constacyclonomials. constacyclonomials cs (x) and cn−s λ If cs (x) does contain such a term βx n−s , it is called a self conjugated constacyclonomial. If csλ (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 csλ∗ (x) of the constacyclonomial csλ (x), s ∈ S n,q,λ , is defined as λ (x) if cλ (x) is not self conjugated, while cλ∗ (x) cλ (x) otherwise. csλ∗ (x) cn−s s s s. 123.

(19) 1210. A. J. van Zanten. From the condition prior to its definition, it follows that csλ∗ (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 An,q,λ :. ⎧ ⎨ ⎩. s∈S. ⎫ ⎬ αs csλ (x)|αs ∈ G F(q) . ⎭ n,q,λ. (10). This set An,q,λ and its elements have the following simple properties. Theorem 15 (i) The polynomial csλ (x)is a constacyclonomial of size m λs if and only if it is λ not the zero polynomial and if m λs is the smallest positive integer satisfying s(q m s −1) 0 mod kn. (ii) Any polynomial p(x) of An,q,λ satisfies p(x)q p(x). m (iii) Let m be the smallest positive integer such that x sq βx s mod x n − λ for some lm−1 β ∈ G F(q)∗ . Then the polynomial p(x) x s + x sq + · · · + x sq , l : or dq (β), is the constacyclonomial csλ (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 csλ (x) are changed into 1, one obtains the cyclonomial cs (x). e,q (vi) By reduction modulo x n − λ of the cyclonomial cs (x), e kn, one obtains either a λ λ. constacyclonomial cs (x), s s mod n, with m s m s , or one obtains the zeropolynomial. (vii) If a constacyclonomial csλ (x) is self conjugated, then either m s 1 and s ∈ {0, n/2}, or m s is even and s(q m s /2 + 1) 0 mod n. Proof. mλ. (i) If the condition in Definition 13 holds we have that x s(q s −1) 1 mod x n − λ. λ When writing s(q m s − 1) an + b, with a ≥ 0, 0 ≤ b < n, it follows that x an+b λ λ λa x b 1. Hence, k |a and b 0. So, kn |s(q m s −1) 0. Conversely, if s(q m s −1) 0 mλ. (ii) (iii). (iv). (v). (vi). mod kn, then x s(q s −1) x ckn λck 1. This statement follows immediately from Definition 13. jm From the given condition it follows that x sq β j x s , for 0 ≤ j ≤ l − 1. If β 1, it follows from Definition 13 that p(x) csλ (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. 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. n,q,λ The first term of both polynomials csλ (x) and cs (x) is x s . Each term of cs (x) is c d c i i i+1 obtained from the previous one ai x by changing it into ai λ x where di [qci /n], n,q while x ci+1 is the next term in cs (x). The statement now follows immediately.. s n It is clear that x mod x − λ is equal to αx s , with s s mod n, for some α ∈ G F(q). e,q e,q Furthermore, it follows from the definition of cs (x) that every term in cs (x) mod. m s x n − λ is equal to the q th power of its predecessor and that αx s x sq mod x n − λ, though m s need not be the smallest integer with this property. The result now follows. 123.

(20) Primitive idempotent tables. 1211. λ (x), then also c (x) c from Definition 13 and part (iii). (vii) If csλ (x) cn−s s n−s (x) by n,q n,q (v), and so the cyclotomic cosets Cs and Cn−s are identical. Hence, sq i and (n − s)q i n,q n,q are both in Cs for all relevant values of i. So, the elements of Cs occur in pairs unless s n − s mod n. 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 cs (x), for fixed values of csλ (x)n, q and λ. It appears that for λ 1 the first number is the smaller one in many cases.. 4.3 A bilinear form in Rn,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 :. n−1 . p(αζ i )q(αζ i ),. (11). i0. 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 Ptλ (x) introduced in Eq. (1) will play a role. We know that the degree of Ptλ (x), t ∈ T n,q,λ , n,q,λ . The coefficient of its one is equal to m λt being the size of the constacyclotomic coset Ct λ −1 n,q,λ m is denoted by pt or shortly by ptλ (remember the conventions but highest power x t mentioned in Sect. 1). Furthermore, we remind the reader of the fact that the size of the n,q constacyclonomial csλ (x), s ∈ S n,q,λ , and also of the cyclotomic coset Cs is equal to m s . n,q,λ In the next theorem and its proof we shall show that the set A of polynomials (10) is an algebra and that the constacyclonomials csλ (x) constitute an orthogonal basis of An,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 An,q,λ is an algebra over G F(q) with basis csλ (x) s ∈ S n,q,λ , and it consists p(x)q p(x). of all polynomials p(x) ∈ R n,q,λ which satisfy the relation n−1 λ j i (ii) For any s ∈ S n,q,λ \{0}, and for any j one has i0 cs (α ζ ) 0, while n−1 λ j i i0 c0 (α ζ ) n. (iii) With respect to the bilinear form (11), the constacyclonomials csλ (x), s ∈ S n,q,λ , form ∗ an orthogonal basis of An,q,λ , such that for any pair j, k ∈ S n,q,λ one has (cλj , ckλ )λ nm j λa j δ j,k , with a j j(q [m j /2] + 1)/n if cλj (x) is self conjugated, while a j 1 if cλj (x) is not self conjugated. Proof. (i) By definition An,q,λ is spanned by the constacyclonomials csλ (x), s ∈ S n,q,λ . All polynomials p(x)∈ An,q,λ satisfy p(x)q p(x) and An,q,λ is a vector space. To see this in detail one should apply the property (βp1 (x) + γ p2 (x))q βp1 (x)q + γ p2 (x)q. 123.

(21) 1212. A. J. van Zanten. 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 An,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 n,q,λ (x) also satisfies p1 (x)q p1 (x), polynomial by csλ (x). Now, p1 (x) : p(x) − βcs n,q,λ n,q,λ and so we can continue this process, leading to p2 (x) : p(x)−βcs (x)−γ cu (x). Proceeding in this way, we finally get the zeropolynomial. We conclude that p(x) n,q,λ n,q,λ (x) + γ cu (x) + · · · ∈ An,q,λ . That An,q,λ is closed under multiplication is a βcs consequence of the relation ( p1 (x) p2 (x))q p1 (x) p2 (x). (ii) If s 0, the polynomial csλ (x) is a sum of terms αl x l , αl ∈ G F(q)∗ , where l runs through a subset U of { 1, 2, . . . , n − 1}. So, any term αl x l occurring in csλ (x) conn−1 λ j i cs (α ζ ) an amount of αl α jl (1 + ζ l + · · · + ζ (n−1)l ), which is tributes to the sum i0 n−1 λ j i c0 (α ζ ) 1 + 1 + · · · + 1 n. equal to zero for all l ∈ U . If s 0, one obtains i0 n,q,λ∗ n,q,λ (iii) Since An,q,λ is a G F(q)-algebra, we may write c j (x)ck (x) n,q,λ n (x) mod x − λ, αs ∈ G F(q). So, the bilinear form (11) yields s∈S n,q,λ αs cs n−1 λ ) (cλ , cλ ) λ λ i , c (cλ∗ λ s∈S n,q,λ i0 αs cs (αζ ). Assume that c j (x) is not self j k n− j k λ λ conjugated. If k j it is obvious that α0 0, and hence by (ii) we have (cλ∗ j , ck )λ 0. λ n− j + x (n− j)q + · · · + x (n− j)q If k j, we have cλ∗ j (x)ck (x) (x. x jq. m j −1. m j −1. )(x j + x jq + · · · +. ) mod x n − λ. Here we used m n− j m j . So, the coefficient of x 0 in the rhs is m j −1. λ m j λ. Hence, the result in this case is (cλ∗ equal to λ+λq +· · ·+λq j , c j )λ nm j λ, n−1 λ i since i0 c0 (αζ ) n because of part (ii) of this theorem. λ λ λ Next, we assume that cλj (x) is self conjugated, and so (cλ∗ j , ck )λ (c j , ck )λ . Like before we may conclude that for k j the rhs is equal to 0, since the inverses of the x-powers in cλj (x) are in cλj (x) itself and not in ckλ (x). Let k j. Since cλj (x) is assumed i. i. to be self conjugated, it contains pairs of terms x jq and x (n− j)q . 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 cλj (x) contains the term λa−1 x n− j λa x − j with a j : j(q m j /2 +1)/n. So, in the product (x j +x jq +· · ·+λa x − j +λaq x − jq +· · ·)(x j +x jq +· · ·+λa x − j +λaq x − jq + m j −1 · · ·), the coefficient of x 0 is equal to λa + λaq + · · · + λaq m j λa , and the result follows in the same way as before. The case j 0, m 0 1 is covered by the general result with a0 0, while for j n/2, m n/2 =1 we have an/2 n2 (q [1/2] + 1)/n 1 which yields also the correct answer. . 5 An orthogonal transformation matrix In this section we shall show that the primitive idempotents θt (x), t ∈ T n,q,λ , form an alternative orthogonal basis for An,q,λ . It then follows that the transformation matrix between this basis and the orthogonal basis of constacyclonomials is an orthogonal matrix.. 123.

(22) Primitive idempotent tables. 1213. 5.1 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 θt (x), t ∈ T n,q,λ , form an orthogonal G F(q)-basis of the vector space An,q,λ , satisfying (θt , θu )λ m λt δt,u for all t ∈ T n,q,λ . (ii) The number n 0 of irreducible polynomials Ptλ (x) and the number of primitive idempotents θt (x), t ∈ T n,q,λ , are both equal to the number of constacyclonomials csλ (x), s ∈ S n,q,λ . Proof (i) From their definition we know θt (x)2 θ (x), and hence θt (x)q θt (x) for all t ∈ T n,q,λ . So, all θt (x) belong to An,q,λ by Theorem 15 (ii). From Theorem 2 (ii) and n,q,λ and equal to 0 otherwise. Theorem 8 (i) it follows that θt (αζ i ) is equal to 1 if i ∈ Ct n−1 i i Hence, (θt , θu )λ i0 θt (αζ )θu (αζ ) is equal to 0 for t u and equal to m λt for t u. To show that An,q,λ is spanned by the idempotent polynomials t ∈ T n,q,λ , n−1 θt (x), n,q,λ i i we assume that p(x) ∈ A is orthogonal to all θt (x). So, i0 θt (αζ ) p(αζ ) 0 for all t ∈ T n,q,λ . Applying Theorem 2 (ii) then yields i∈C n,q,λ p(αζ i ) 0, t ∈ T n,q,λ . t From Theorem 15 (ii) we have that p(x)q p(x), or equivalently p(x q ) p(x). Since a n,q,λ there is a positive integer a such that αζ i (αζ j )q , we can for any pair i, j ∈ Ct n,q,λ i λ j and for all t ∈ T n,q,λ . It write i∈C n,q,λ p(αζ ) m t p(αζ ) for some j ∈ Ct t. 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 θt (x) form an orthogonal basis of An,q,λ . (ii) Since the basis of primitive idempotents θt (x), t ∈ T n,q,λ , and the basis of constacyλ n,q,λ clonomials. cs (x),. s ∈ S , 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 θt (x) and the irreducible polynomial Ptλ (x) with zeros αζ t for t ∈ T n,q,λ . Since the constacyclonomials constitute an orthogonal basis for An,q,λ , each element p ∈ An,q,λ can be developed as (cf. Theorem 17 (iii)) ξs csλ (x), ξs (csλ∗ , p)λ /nm s λas . (12) p(x) s∈S n,q,λ. In particular we can write for the primitive idempotent θt (x), t ∈ T λ , the expression ξst csλ (x), ξst (csλ∗ , θt )λ /nm s λas . θt (x) . (13). s∈S n,q,λ. Theorem 19 (Orthogonality relations for primitive idempotents) (i) Let μs,t stand for the sum of the s-powers of the zeros of Ptλ (x), for s ∈ S n,q,λ and t ∈ T n,q,λ . Then the coefficients of the idempotent θt (x) can be written as ξst μs,t /nλas when csλ (x) is self conjugated, and as ξst μn−s,t /nλas when csλ (x)is not self conjugated. (ii) The sum μs,t can be written in terms of irreducible polynomials as μs,t λs /m λs −m λt p(k,s)t (k,s)t . ws t u ws t t (iii) If ws : m s λas , then n s∈S n,q,λ m λ ξs ξn−s δt,u and n t∈T n,q,λ m λ ξs ξn−r δs,r for t, u ∈ T n,q,λ and s, r ∈ S n,q,λ .. t. t. 123.

(23) 1214. Proof. A. J. van Zanten. (i) We know that θtλ (αζ i ) is equal to 1 if αζ i is a zero of the irreducible polynomial Ptλ (x), while it is equal to 0 otherwise. Let csλ (x) be self conjugated. 2 Then csλ∗ (αζ i ) csλ (αζ i )=(αζ i )s + (αζ i )sq + (αζ i )sq · · ·, and hence (csλ∗ , θtλ )λ n−1 λ i λ i i s i sq i sq 2 + · · ·, where the sumi (αζ ) + i (αζ ) + i (αζ ) i0 cs (αζ )θt (αζ ) n,q,λ . Since αζ i mation indices i in the m s terms of the rhs run through the set Ct j λ i q λ is a zero of Pt (x), we have that (αζ ) is also a zero of Pt (x) and we can write j n,q,λ (αζ i )q αζ i , i ∈ Ct .The mapping i → i is one-to-one, and so the m s summations are all equal to i∈C n,q,λ (αζ i )s . The individual terms in this summation are t. the s -powers of the zeros of Ptλ (x), and they are zeros themselves of some irreducible s polynomial Ptλs (x). So, ξst m s μs,t /nm s λas = μs,t /nλas . If csλ (x) is not self conλ (x) . The result then follows in a similar way and by jugated, we have csλ∗ (x) cn−s using an−s as 1 in this case. (ii) This follows immediately from Theorem 10 (v). t u λ λ (iii) Substituting the given expressions yields (θt , θu )λ s,s ξs ξs (cs , cs )λ = u u t λ λ a t s s∈S n,q,λ ξs ξn−s (cs , cn−s )λ n s∈S n,q,λ m s λ ξs ξn−s , by applying Theorem 17 (iii). By putting ws m s λas the first relation follows. Combining the expression for ξst in λ (αζ t ). (i) with the expression in Theorem 17 (iii) provides us with nws ξst m λt cn−s n−1 λ 1 t t 2 i λ i Hence, n ws wr t∈T n,q,λ m λ ξs ξr i0 cn−s (αζ )cn−r (αζ ) nws δs,n−r , again t by applying Theorem 17 (iii). The second relation follows by replacing r by n − r . 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 λt 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 w ξt m s μs,t equality which makes sense. E.g. the fraction ms λs nm in (iii) appears to be a well λ t t 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]. 5.2 Idempotent tables Ξ n,q,λ and Mn,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 0 n,q,λ : ξst , s ∈ S n,q,λ , t ∈ T n,q,λ . The adjoint matrix Ξ n,q,λ over G F(q) has elements Ξs,t n,q,λ∗ ws t : m matrix Ξ n,q,λ∗ is the matrix with elements Ξs,t λ ξn−s . t. Theorem 21 (i) Ξ n,q,λ Ξ n,q,λ∗ Ξ n,q,λ∗ Ξ n,q,λ n I . (ii) Let e(x) t∈T n,q,λ ηt θt (x), be the idempotent generator of some constacyclic code C n,q,λ , and let η (η0 , η1 , . . . , ηn 0 −1 )T ∈ G F(q)n 0 . Then e(x) s∈S n,q,λ ξs csλ (x), ξ (ξ0 , ξ1 , . . . , ξn 0 −1 )T n −1 Ξ n,q,λ∗ η. The next simple example will enable the reader to verify all properties stated in Theorems 18 and 19. Example 22 Let n 12, q 7 and λ 2. Hence, k 3 and l 2.. 123.

(24) Primitive idempotent tables. 1215. The binomial x 12 − 2 can be factorized into three irreducible polynomials over G F(7) as (x 6 − 3)(x 3 + 2)(x 3 − 2). Let α be a zero of x 6 − 3 of order 36. The zeros of x 12 − 2 can be written as αζ i , 0 ≤ i < 12, with ζ α 3 . The three constacyclonomials in this case are C012,7,2 (0, 2, 4, 6, 8, 10), C112,7,2 (1, 9, 5) and C312,7,2 (3, 11, 7). Since α 12 2, we have (αζ 1 )3 α 12 2 and (αζ 3 )3 α 30 5, and so we can index the irreducible factors as P02 (x) x 6 − 3, P12 (x) x 3 − 2 and P32 (x) x 3 + 2. Though there are nine cyclotomic cosets Cs12,7 in this case, only three of them, i.e. C012,7 (0), C312,7 (3, 9) and C612,7 (6), give rise to a constacyclonomial cs2 (x) : cs12,7,2 (x), according to Theorem 18 (ii). These three constacyclonomials are c02 (x) 1, c32 (x) x 3 + 2x 9 and c612 (x) x 6 , and we define S 12,7,2 : {0, 3, 6}. There are also three primitive idempotents θt (x), t ∈ T 12,7,2 , with T 12,7,2 {0, 1, 3}. By applying the general expression of Theorem 3 (ii), we find θ0 (x) −x 6 + 4, θ1 (x) 2x 9 + 4x 6 + x 3 + 2 and θ3 (x) −2x 9 + 4x 6 − x 3 + 2. These expressions, together with (13), yield the transformation matrix ⎤ ⎡ 422 Ξ 12,7,2 ⎣ 0 1 6 ⎦. 644 x 12 − 2. The rows are indexed by respectively 0, 3, 6, the integers of S 12,7,2 , and the columns by 0, 1, 3, the integers of T 12,7,2 . We collect the weights ws ( m s λas ), s ∈ S 12,7,2 , in the weight vector σ (1, 1, 2) ∈ G F(7)3 , and similarly the weights 1/m t , t ∈ T 12,7,2 , in the weight vector τ (6, 5, 5) ∈ G F(7)3 . With the help of these vectors, one can easily verify in this case the orthogonality relations of Theorem 19 (iii). We also present the closely related matrix M12,7,2 , the elements μs,t of which are the sums of the s-powers of the zeros of the irreducible polynomials Pt2 (x). Since the three constacyclonomials are self conjugated, M12,7,2 is obtained by multiplying the rows of Ξ 12,7,2 by nλas for s equal to 0, 3 and 6. With a0 0, a3 2 and a6 1, we get ⎡ ⎤ 633 M12,7,2 ⎣ 0 6 1 ⎦. 455 One can also produce this result by applying Theorems 19 (ii) and 10 (v), or by determining the sums μs,t straightforwardly. Apart from the polynomials Pt2 (x), the relevant irreducible 0 3 6 0 polynomials to carry this out are P02 (x) x − 1, P02 (x) x 2 + 4, P02 (x) x − 3, P32 3 6 0 3 6 x + 1, P32 (x) x − 2, P32 (x) x + 3, P92 (x) x + 1, P92 (x) x + 2 and P92 (x) x + 3. 0 3 Notice that e.g. P02 (x) and P02 (x), though 20 23 1 in G F(7), are different polynomials, due to the representation of their zeros as defined in Theorem 10 (v). 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 , n,q,λ , correspond to irreducible characters. being the indices of the constacyclotomic cosets Ct n,q n,q,λ The labels s ∈ S of the rows are the indices of those cyclotomic cosets Cs 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 Mn,q,λ (e.g. in Example 22) with elements n,q,λ. μs,t nλas Ξs∗,t ,. (14). 123.

(25) 1216. A. J. van Zanten. where s ∗ s when csλ (x) is self conjugated, and s ∗ n − s otherwise (cf. Theorem 19 (i)).. 6 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.. 6.1 Primitive idempotent tables Mn,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 n,q n,q more similarities with character tables, we defined in Sect. 4 Ct ∗ : C−t as the conjugated n,q n,q ∗ cyclotomic coset of Ct , t ∈ T , where t : n − t is an integer in [0, n − 1]. It will be obvious that m t ∗ m t . Correspondingly, we define Pt ∗ (x) of degree m t as the conjugated irreducible polynomial of Pt (x) and θt ∗ (x) as the conjugated primitive idempotent of θt (x). Actually, the polynomial Pt ∗ (x) is the monic reciprocal of Pt (x), formally expressed by Pt ∗ (x) Pt (0)−1 x m t Pt (1/x),. (15). which was introduced in Theorem 2 (vi) for any Ptλ (x), t ∈ T n,q,λ . We say that Pt (x) is self conjugated if Pt ∗ (x) Pt (x), and similarly that θt (x) is self conjugated if θt ∗ (x) θt (x). We also introduced in Definition 14 the constacyclonomial csλ∗ (x) as the conjugate of csλ (x) . λ (x), so s∗ s if C λ (x) is self conjugated and s∗ n − s We now write this polynomial as cs∗ 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 θt (x) for a cyclic code can be written as ξst μs,t /n, where μs,t stands for the sum of the s-powers of the zeros of Pt (x), for all s ∈ S n,q and t ∈ T n,q . (ii) The sum μs,t can be expressed in terms of irreducible polynomials as μs,t −m t pst /m st . n,q n,q t ξ t∗ . (iii) For has ξs∗ s all s, rm∈s St u and for all t, u, ∈ T m, sone t t (iv) n s∈S n,q m t ξs ξs∗ δt,u and n t∈T n,q m t ξs ξr ∗ δs,r . (v) The number of self conjugated primitive idempotent generators is equal to the number of self conjugated cyclonomials. The expression in (ii) is immediately clear if one realizes that the s-powers of the zeros of Pt (x) are zeros of Pst (x). This expression was already derived in [25] in a slightly different way. The equality in (iii) follows from (i) and (ii) and by applying m n−s m s and m n−t m t . The statement in (v) is also obvious, since for all s ∈ S n,q ( T n,q ) both polynomials cs (x) n,q and θs (x) are self conjugated if and only if the corresponding cyclotomic coset Cs is self conjugated. Theorem 24 (Primitive idempotent table for cyclic codes) The entriesμs,t of the table Mn,q , s ∈ S n,q , t ∈ T n,q ( S n,q ), satisfy the following properties.. 123.

(26) Primitive idempotent tables. 1217. ms ms (i) s∈S n,q m t μs,t μs∗,u nδt,u , t∈T n,q m t μs,t μr ∗,t nδs,r , (ii) μs∗,t μs,t∗ , m s μs,t m t μt,s . (iii) μs,0 1 for all s ∈ S n,q and μ0,t m t for all t ∈ T n,q . (iv) If n is even μs,n/2 (−1)s for all s ∈ S n,q and μn/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 μs,t in Theorem 19 (ii). (iii) and (iv) follow from Theorem 19 (ii) by substituting respectively p0 −1 and pn/2 1. These values are yielded by the irreducible polynomials P0 (x) x − 1 and Pn/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 Mn,q with index 0 (its ‘first’ column) is the all-one column, and so θ0 corresponds to the trivial character χ1 of a finite group G. Furthermore, the row of Mn,q with index 0 (its ‘first’ row) contains all values m t , i.e. the sizes of the j cyclonomials. One could consider these values as counterparts of the dimensions (degrees) χ1 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 j 2 equality as the counterpart of the well known Burnside relation j (χ1 ) 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 Mn,q , and more in general for Mn,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]).. 6.2 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 Un consisting of the positive integers modulo n which are prime n,q n,q to n. It will be obvious that the set rCs is identical to the cyclotomic coset Cr s . We n,q shall call it the r -conjugate of Cs . Similarly, the cyclonomial cr s (x) is the r -conjugate of cs (x), and the irreducible polynomial Pr t (x) the r -conjugate of Pt (x). Since (n, r ) 1, one easily proves that m r s m s , that cr s (x) and cs (x) have the same size, and that Pr t (x) and Pt (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 cs (x) is r -self conjugated if cr s (x) cs (x), and that Pt (x) is r -self conjugated if Pr t (x) Pt (x). If θr t (x) is the primitive idempotent generated by Pr t (x), then θr t (x) and θt (x) are also said to be r-conjugated, and if they are equal θt (x) is said to be r-self conjugated. There exists a simple relationship between the primitive idempotent θt (x) and its r-conjugate θr t (x). Let 1/r be the inverse of r in Un . From Theorem 2 (ii) it follows that θt (x 1/r ) 1 for x β r and β is a zero of Pt (x), and that θt (x 1/r ) 0 for x β r and β is a zero of Pu (x), u t. We conclude that for all t ∈ T n,q θr t (x) θt (x 1/r ), r ∈ Un .. (16). 123.