Cyclotomic Numbers Research Articles

Upper Bounds for Cyclotomic Numbers

Let q be a power of a prime p, let k be a nontrivial divisor of q-1 and write e=(q-1)/k. We study upper bounds for cyclotomic numbers (a,b) of order e over the finite field 𝔽 q . A general result of our study is that (a,b)≤3 for all a,b∈ℤ if p>(14) k/ord k (p) . More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: (0,0),(0,a),(a,0),(a,a) and (a,b), where a≠b and a,b∈{1,...,e-1}. The main idea we use is to transform equations over 𝔽 q into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers.

Linear Independence of Logarithms of Cyclotomic Numbers and a Conjecture of Livingston

AbstractIn 1965, A. Livingston conjectured the $\overline{\mathbb{Q}}$-linear independence of logarithms of values of the sine function at rational arguments. In 2016, S. Pathak disproved the conjecture. In this article, we give a new proof of Livingston’s conjecture using some fundamental trigonometric identities. Moreover, we show that a stronger version of her theorem is true. In fact, we modify this conjecture by introducing a co-primality condition, and in that case we provide the necessary and sufficient conditions for the conjecture to be true. Finally, we identify a maximal linearly independent subset of the numbers considered in Livingston’s conjecture.

Complete solution to cyclotomy of order $$2l^{2}$$ with prime l

Let l, p be odd primes, $$q=p^r,$$ $$r\in \mathbf {Z}^{+},$$ $$q\equiv 1\pmod {2l^2}$$ and $$\mathbf {F}_q$$ a field with q elements. The problem of determining cyclotomic numbers in terms of the solutions of certain Diophantine systems has been treated by many authors since the age of Gauss but still effective formulae are yet to be known. In this paper we obtain an explicit expression for cyclotomic numbers of order $$2l^{2}.$$ The formula consists of the cyclotomic numbers of orders $$\displaystyle {l, 2l, l^2}$$ and the coefficients of a special type Jacobi sum of order $$2l^2.$$ At the end, we illustrate the nature of two matrices corresponding to two types of cyclotomic numbers.

Generalized cyclotomic numbers and cyclic codes of prime power length over Z4

Generalized cyclotomic numbers of order [Formula: see text] with respect to an odd prime power are obtained. Hence, explicit expressions for primitive idempotents in the ring [Formula: see text] are obtained in two cases, when the multiplicative order of 2 modulo [Formula: see text] is [Formula: see text] and [Formula: see text], where [Formula: see text] is an odd prime. Orthogonality and self-duality of some [Formula: see text] cyclic codes are also discussed. Further, a method for obtaining cyclic self-dual/isodual codes of length [Formula: see text] over [Formula: see text] is given.

A new family of optimal FHS sets with composite lengths

In frequency-hopping spread spectrum systems, frequency-hopping sequences (FHSs) are widely used to alleviate the interferences caused by the hits of frequencies. In this paper, based on the Zeng–Cai–Tang–Yang cyclotomy and the Chinese Remainder Theorem, a family of FHS sets with composite lengths is constructed and the Hamming correlations of the new FHS sets are derived by some basic properties of cyclotomic numbers. The results show that the proposed FHS sets are optimal with respect to the Peng–Fan bound. Furthermore, it generalizes some earlier cyclotomic constructions of optimal FHS sets and produces optimal FHS sets with new parameters which are not covered in the literature.

Linear complexity of Ding-Helleseth generalized cyclotomic sequences of order eight

During the last two decades, many kinds of periodic sequences with good pseudo-random properties have been constructed from classical and generalized cyclotomic classes, and used as keystreams for stream ciphers and secure communications. Among them are a family DH-GCS$_{d}$ of generalized cyclotomic sequences on the basis of Ding and Helleseth's generalized cyclotomy, of length $pq$ and order $d=\mathrm{gcd}(p-1,q-1)$ for distinct odd primes $p$ and $q$. The linear complexity (or linear span), as a valuable measure of unpredictability, is precisely determined for DH-GCS$_{8}$ in this paper. Our approach is based on Edemskiy and Antonova's computation method with the help of explicit expressions of Gaussian classical cyclotomic numbers of order $8$. Our result for $d=8$ is compatible with Yan's low bound $(pq-1)/2$ of the linear complexity for any order $d$, which means high enough to resist security attacks of the Berlekamp-Massey algorithm. Finally, we include SageMath codes to illustrate the validity of our result by examples.

Computation of jacobi sums and cyclotomic numbers with reduced complexity

Jacobi sums and cyclotomic numbers are the important objects in number theory. The determination of all the Jacobi sums and cyclotomic numbers of order $e$ are merely intricate to compute. This paper presents the lesser numbers of Jacobi sums and cyclotomic numbers which are enough for the determination of all Jacobi sums and the cyclotomic numbers of a particular order.

Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$

Let $K$ be a real abelian extension of $\mathbb{Q}$. Let $p$ be a prime number, $S$ the set of $p$-places of $K$ and ${\mathcal G}_{K,S}$ the Galois group of the maximal $S \cup \{\infty\}$-ramified pro-$p$-extension of $K$ (i.e., unramified outside $p$ and $\infty$). We revisit the problem of annihilation of the $p$-torsion group ${\mathcal T}_K := \text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ initiated by us and Oriat then systematized in our paper on the construction of $p$-adic $L$-functions in which we obtained a canonical ideal annihilator of ${\mathcal T}_K$ in full generality (1978--1981). Afterwards (1992--2014) some annihilators, using cyclotomic units, were proposed by Solomon, Belliard--Nguyen Quang Do, Nguyen Quang Do--Nicolas, All, Belliard--Martin. In this text, we improve our original papers and show that, in general, the Solomon elements are not optimal and/or partly degenerated. We obtain, whatever $K$ and $p$, an universal non-degenerated annihilator in terms of $p$-adic logarithms of cyclotomic numbers related to $L_p$-functions at $s=1$ of {primitive characters of $K$} (Theorem 9.4). Some computations are given with PARI programs; the case $p=2$ is analyzed and illustrated in degrees $2$, $3$, $4$ to test a conjecture.

Non-existence of Some Nearly Perfect Sequences, Near Butson–Hadamard Matrices, and Near Conference Matrices

In this paper we study the non-existence problem of (nearly) perfect (almost) m-ary sequences via their connection to (near) Butson–Hadamard (BH) matrices and (near) conference matrices. We refine the idea of Brock on the unsolvability of certain equations in the case of cyclotomic number fields whose ring of integers is not a principal ideal domain and get many new non-existence results for near BH matrices and near conference matrices. We also apply previous results on vanishing sums of roots of unity and self conjugacy condition to derive non-existence results for near BH matrices and near conference matrices.

Generalized cyclotomic numbers with respect to prime powers

In this paper, we study the generalized cyclotomic numbers with respect to the powers of odd primes, where at most one prime is of the form [Formula: see text]. We calculate the generalized cyclotomic numbers of order [Formula: see text] with respect to [Formula: see text] when the odd primes [Formula: see text] and the positive integers [Formula: see text] satisfy [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text].

De Bruijn Sequences, Adjacency Graphs, and Cyclotomy

We study the problem of constructing De Bruijn sequences by joining cycles of linear feedback shift registers (LFSRs) with reducible characteristic polynomials. The main difficulty for joining cycles is to find the location of conjugate pairs between cycles, and the distribution of conjugate pairs in cycles is defined to be adjacency graphs. Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l(x)$ </tex-math></inline-formula> be a characteristic polynomial, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l(x)=l_{1}(x)l_{2}(x)\cdots l_{r}(x)$ </tex-math></inline-formula> be a decomposition of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l(x)$ </tex-math></inline-formula> into pairwise co-prime factors. First, we show a connection between the adjacency graph of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {FSR}(l(x))$ </tex-math></inline-formula> and the association graphs of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {FSR}(l_{i}(x))$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1\leq i\leq r$ </tex-math></inline-formula> . By this connection, the problem of determining the adjacency graph of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {FSR}(l(x))$ </tex-math></inline-formula> is decomposed to the problem of determining the association graphs of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {FSR}(l_{i}(x))$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1\leq i\leq r$ </tex-math></inline-formula> , which is much easier to handle. Then, we study the association graphs of LFSRs with irreducible characteristic polynomials and give a relationship between these association graphs and the cyclotomic numbers over finite fields. At last, as an application of these results, we explicitly determine the adjacency graphs of some LFSRs and show that our results cover the previous ones.

Music as Mathematics of Senses

It is often said that music has reached its supreme and highest level in the 18th and 19th centuries. One of the main reasons for this achievement seems to be the robust structure of compositions of music, somewhat remindful of robust structure of mathematics. One is reminded of the words of Goethe: Geometry is frozen music. Here, we may extend geometry to mathematics. For the Middle Age in Europe, there were seven main subjects in the universities or in higher education. They were grammar, logic and rhetoric—these three (tri) were regarded as more standard and called trivia (trivium), the origin of the word trivial. And the remaining four were arithmetic, geometry, astronomy and music—these four (quadrus) were regarded as more advanced subjects and were called quadrivia (quadrivium). Thus for Goethe, geometry and mathematics seem to be equivocal. G. Leibniz expresses more in detail in his letter to C. Goldbach in 1712 (April 17): Musica est exercitium arithmeticae occultum nescientis se numerari animi (Music is a hidden arithmetic exercise of the soul, which doesn’t know that it is counting). Or in other respects, J. Sylvester expresses more in detail: Music is mathematics of senses. Mathematics is music of reasons. Thus, the title arises. This paper is a sequel to [1] and examines mathematical structure of musical scales entailing their harmony on expanding and elaborating material in [2] [3] [4] [5], etc. In statistics, the strong law of large numbers is well-known which claims that This means that the relative frequency of occurrences of an event A tends to the true probability p of the occurrences of A with probability 1. In music, harmony is achieved according to Pythagoras’ law of small numbers, which claims that only the small integer multiples of the fundamental notes can create harmony and consonance. We shall also mention the law of cyclotomic numbers according to Coxeter, which elaborates Pythagoras’ law and suggests a connection with construction of n-gons by ruler and compass. In the case of natural scales (just intonation), musical notes appear in the form 2p3q5r (multiples of the basic note), where p∈Z, q=-3, -2, -1, 0, 1, 2, 3 and r=-1, 0, 1. We shall give mathematical details of the structure of various scales.

A class of cyclic codes constructed via semiprimitive two-weight irreducible cyclic codes

We present a family of reducible cyclic codes constructed as a direct sum (as vector spaces) of two different semiprimitive two-weight irreducible cyclic codes. This family generalizes the class of reducible cyclic codes that was reported in the main result of [10]. Moreover, despite of what was stated therein, we show that, at least for the codes studied here, it is still possible to compute the frequencies of their weight distributions through the cyclotomic numbers in an easy way.

A construction of linear codes and their complete weight enumerators

Recently, linear codes constructed from defining sets have been studied extensively. They may have excellent parameters if the defining set is chosen properly. Let m>2 be a positive integer. For an odd prime p, let r=pm and Tr be the absolute trace function from Fr onto Fp. In this paper, we give a construction of linear codes by defining the codeCD={(Tr(ax))x∈D:a∈Fr}, where D={x∈Fr:Tr(x)=1,Tr(x2)=0}. Its complete weight enumerator and weight enumerator are determined explicitly by employing cyclotomic numbers and Gauss sums. However, we find that the code is optimal with respect to the Griesmer bound provided that m=3. In fact, it is MDS when m=3. Moreover, the codes presented have higher rate compared with other codes, which enables them to have essential applications in areas such as association schemes and secret sharing schemes.

Two classes of near-optimal frequency-hopping sequence sets with prime-power period

In this paper, a kind of generalized cyclotomy with respect to a prime power is introduced and properties of the corresponding generalized cyclotomic numbers are investigated. Based on the generalized cyclotomy, two classes of frequency-hopping sequence (FHS) sets with prime-power period are presented. Meanwhile, we derive the Hamming correlation distribution of the new FHS sets. The results show that the proposed FHSs and FHS sets have (near-) optimal maximum Hamming correlation (MHC). These classes of near-optimal FHS sets have new parameters which are not covered in the literature.

Nonexistence of twenty-fourth power residue addition sets

Let n>1 be an integer, and let Fp denote a field of p elements for a prime p≡1(modn). By 2015, the question of existence or nonexistence of n-th power residue difference sets in Fp had been settled for all n<24. We settle the case n=24 by proving the nonexistence of 24-th power residue difference sets in Fp. We also prove the nonexistence of qualified 24-th power residue difference sets in Fp. The proofs make use of a Mathematica program which computes formulas for the cyclotomic numbers of order 24 in terms of parameters occurring in quadratic partitions of p.

完全数,円分数,及びABC予想

完全数,円分数,及びABC予想

Construction of de Bruijn sequences from product of two irreducible polynomials

We study a class of Linear Feedback Shift Registers (LFSRs) with characteristic polynomial $f(x)=p(x)q(x)$ where $p(x)$ and $q(x)$ are distinct irreducible polynomials in $\F_2[x]$. Important properties of the LFSRs, such as the cycle structure and the adjacency graph, are derived. A method to determine a state belonging to each cycle and a generic algorithm to find all conjugate pairs shared by any pair of cycles are given. The process explicitly determines the edges and their labels in the adjacency graph. The results are then combined with the cycle joining method to efficiently construct a new class of de Bruijn sequences. An estimate of the number of resulting sequences is given. In some cases, using cyclotomic numbers, we can determine the number exactly.

The Adjacency Graphs of LFSRs With Primitive-Like Characteristic Polynomials

We consider the adjacency graphs of the linear feedback shift registers (LFSRs) with characteristic polynomials of the form $l(x)p(x)$ , where $l(x)$ is a polynomial of small degree and $p(x)$ is a primitive polynomial. It is shown that their adjacency graphs are closely related to the association graph of $l(x)$ and the cyclotomic numbers over finite fields. By using this connection, we give a unified method to determine their adjacency graphs. As an application of the method, we explicitly calculate the adjacency graphs of LFSRs with the characteristic polynomials of the form $(1+x+x^{3}+x^{4})p(x)$ , and construct a large class of De Bruijn sequences from them.

Galois module structure of the square root of the inverse different in even degree tame extensions of number fields

Let G be a finite group and let N/E be a tamely ramified G-Galois extension of number fields whose inverse different CN/E is a square. Let AN/E denote the square root of CN/E. Then AN/E is a locally free Z[G]-module, which is in fact free provided N/E has odd order, as shown by Erez. Using M. Taylor's theorem, we can rephrase this result by saying that, when N/E has odd degree, the classes of AN/E and ON (the ring of integers of N) in Cl(Z[G]) are equal (and in fact both trivial). We show that the above equality of classes still holds when N/E has even order, assuming that N/E is locally abelian. This result is obtained through the study of the Fröhlich representatives of the classes of some torsion modules, which are independently introduced in the setting of cyclotomic number fields. Jacobi sums, together with the Hasse–Davenport formula, are involved in this study. Finally, when G is the binary tetrahedral group, we use our result in conjunction with Taylor's theorem to exhibit a tame G-Galois extension whose square root of the inverse different has nontrivial class in Cl(Z[G]).