Cyclotomic Classes Research Articles

Almost Difference Sets from Unions of Cyclotomic Classes of Order 10

This study investigated the problem of constructing almost difference sets from single and unions of cyclotomic classes of order 10 (with and without zero) of the finite field GF(q), where q is a prime of the form q = 10n+1 for integer n≥1 and q<1000. Using exhaustive computer searches in Python, the method computed unions of two classes up to nine cyclotomic classes. Moreover, the equivalence of the generated almost difference sets having the same parameters was determined up to complementation. Results showed that a single cyclotomic class formed an almost difference set for q = 31, 71, and 151, while including zero produced an almost difference set for q=11, 31, 71, and 151. Additionally, Paley partial difference sets were constructed from the union of five cyclotomic classes. These findings addressed gaps in the literature on cyclotomy of order 10.

SYMMETRIC 2-ADIC COMPLEXITY OF GENERALIZED CYCLOTOMIC SEQUENCES WITH PERIOD 𝟐𝒑 �

Abstract. In this paper, we study the symmetric 2-adic complexity of generalized cyclotomic sequences with period 2𝑝 𝑛. These sequences are based on generalized binary cyclotomic classes of order two and have high linear complexity. The 2-adic complexity is another measure of the predictability of a sequence and thus its unsuitability for cryptography. We prove that the symmetric 2-adic complexity of considered sequences attains the maximal value. The generalized “Gauss periods” are used to derive 2-adic complexity of these sequences

On Some New Almost Difference Sets Constructed from Cyclotomic Classes of Order 12

Almost Difference Sets have extensive applications in coding theory and cryptography. In this study, we introduce new constructions of Almost Difference Sets derived from cyclotomic classes of order 12 in the finite field GF(q), where q is a prime satisfying the form q=12n+1 for positive integers n ≥ 1 and q < 1000. We show that a single cyclotomic class of order 12 (with and without zero) can form an almost difference set. Additionally, we successfully construct almost difference sets using unions of cyclotomic classes of order 12, both for even and odd values of n. To accomplish this, an exhaustive computer search employing Python was conducted. The method involved computing unions of two cyclotomic classes up to eleven classes and assessing the presence of almost difference sets. Finally, we classify the resulting almost difference sets with the same parameters up to equivalence and complementation.

Difference sets in pseudocyclic association schemes

Wilson (1972) proved that for fixed positive integers d and r such that 2d divides r(r−1), there exists a subset Y of size r of the finite field of order q such that the r(r−1) nonzero differences occurring from Y are evenly distributed to the dth cyclotomic classes provided that q is sufficiently large relative to d and r. On the other hand, Kajiura, Matsumoto and Okuda (2019) introduced the concept of difference sets in commutative association schemes in their study on a quasi-Monte Carlo method to approximate integrations over the point sets. Then, Wilson's result implies the existence of difference sets in cyclotomic schemes. In this paper, we generalize Wilson's result into pseudocyclic commutative association schemes and show the existence of difference sets in pseudocyclic association schemes with sufficiently large numbers of points.

Lee weight distributions of trace codes over F_q+uF_q from irreducible cyclic codes

The construction of two or few-weight codes from trace codes over the ring Fq + uFq, where u2 = 0, was recently presented in [4]. For such construction, the defining sets for the trace codes are given in terms of cyclotomic classes, and for some of these classes, it is shown that it is possible to obtain the Lee weight distributions of the corresponding trace codes. Motivated by this construction, and by the p-ary semiprimitive irreducible cyclic codes over a prime field Fp, the Lee weight distributions of an infinite family of p-ary three-weight codes from trace codes over the ring Fq + uFq, was recentely found in [11]. In this work, we prove that the Lee weight distribution problem for the trace codes constructed in accordance with either [4] or [11], is equivalent to the weight distribution problem for the irreducible cyclic codes. With this equivalence in mind, and by using the already known weight distributions of an infinite family irreducible cyclic codes (semiprimitive and not semiprimitive), we follow the open problem suggested in the Conclusion of [11] to determine the Lee weight distribution of an infinite family of trace codes over the ring Fq + uFq , that includes the infinite family found in [11].

On the differential spectrum of a differentially 3-uniform power function

In this paper, we investigate the cardinality of the intersection of four particular cyclotomic classes of order two over the finite field with characteristic 5 by making most use of the cyclotomic numbers of orders two and four. As a consequence, a class of power functions over F5n is shown to be differentially 3-uniform and its differential spectrum is also completely determined.

The EKR-Module Property of Pseudo-Paley Graphs of Square Order

We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erdős-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear combination of characteristic vectors of canonical cliques. This extends the EKR-module property of Paley graphs of square order and solves a problem proposed by Godsil and Meagher. Different from previous works, which heavily rely on tools from number theory, our approach is purely combinatorial in nature. The main strategy is to view these graphs as block graphs of orthogonal arrays, which is of independent interest.

An extension of binary cyclotomic sequences having order 2lt

Several reasonably cyclotomic sequences are constructed by cyclotomic classes having good pseudo-randomness property. In this paper, we derive the linear complexity of an extended binary cyclotomic sequences of order [Formula: see text] over finite field having period [Formula: see text]. Our result shows that these sequences have higher linear complexity, which can resist linear attack.

Constructing few-weight linear codes and strongly regular graphs

Linear codes with few weights have applications in data storage systems, secret sharing schemes, graph theory and so on. In this paper, we construct a class of few-weight linear codes by choosing defining sets from cyclotomic classes and we also establish few-weight linear codes by employing weakly regular bent functions. Notably, we get some codes that are minimal and we also obtain a class of two-weight optimal punctured codes with respect to the Griesmer bound. Finally, we get a class of strongly regular graphs with new parameters by using the obtained two-weight linear codes.

Cyclotomy, cyclotomic cosets and arithmetic properties of some families in 𝔽l[x] 〈xptq−1〉

Arithmetic properties of some families in [Formula: see text] and hence in [Formula: see text] are obtained by using the cyclotomic classes of order 2 with respect to [Formula: see text], where [Formula: see text] is primitive root modulo [Formula: see text], [Formula: see text] and [Formula: see text]. The form of these cyclotomic classes enables us to generalize the results obtained in [S. Jain and S. Batra, Cyclotomy and arithmetic properties of some families in [Formula: see text], Asian-Eur. J. Math. 13(4) (2020) 2050077].

The punctured codes of two classes of cyclic codes with few weights

<p style='text-indent:20px;'>The puncturing technique is one of important tools for constructing new codes from old ones. In recent years, many classes of linear codes with interesting parameters have been obtained with this technique. Based on quadratic Gauss sums, the puncturing technique and cyclotomic classes, we investigate two classes of punctured codes of cyclic codes over finite fields. The weight distribution and the parameters of the duals of these codes are studied. The parameters of these codes are new.</p>

On the $ k $-error linear complexity of binary sequences of periods $ p^n $ from new cyclotomy

<abstract><p>In this paper, we study the $ k $-error linear complexity of binary sequences with periods $ p^n $, which are derived from new generalized cyclotomic classes modulo a power of an odd prime $ p $. We establish a recursive relation and then estimate the $ k $-error linear complexity of the binary sequences with periods $ p^n $, the results extend the case $ p^2 $ that has been studied in an earlier work of Wu et al. at 2019. Our results show that the $ k $-error linear complexity of these sequences does not decrease dramatically for $ k < (p^{n}-p^{n-1})/2 $.</p></abstract>

Difference Sets from Unions of Cyclotomic Classes of Orders 12, 20, and 24

Let 𝒒𝒒 be a prime of the form 𝒒𝒒 = 𝒏𝒏𝒏𝒏 + 𝟏𝟏 for integers 𝒏𝒏 ≥ 𝟏𝟏 and 𝑵𝑵 > 𝟏𝟏. For 𝒒𝒒 < 𝟏𝟏𝟏𝟏𝟓𝟓, we show that difference sets in the additive group of the field 𝐆𝐆𝑭𝑭(𝒒𝒒) are obtained from unions of cyclotomic classes of orders 𝑵𝑵 = 𝟏𝟏𝟏𝟏, 𝟐𝟐𝟐𝟐, and 𝟐𝟐𝟐𝟐 and determine all such unions using a computer search. We then determine if the difference sets are equivalent to known cyclotomic or modified cyclotomic quadratic, quartic, sextic, or octic difference sets or their complements. This fills the gaps in the literature on the existence of difference sets from unions of cyclotomic classes for the specified orders. In addition, we extend Baumert and Fredricksen’s 1967 work on the construction of all inequivalent (𝟏𝟏𝟏𝟏𝟏𝟏, 𝟔𝟔𝟔𝟔, 𝟑𝟑𝟑𝟑)-difference sets from unions of 𝟏𝟏𝟏𝟏𝒕𝒕𝒕𝒕- cyclotomic classes of 𝑮𝑮𝑮𝑮(𝟏𝟏𝟏𝟏𝟏𝟏) by constructing six inequivalent (𝟏𝟏𝟏𝟏𝟏𝟏, 𝟔𝟔𝟔𝟔, 𝟑𝟑𝟑𝟑)-difference sets with zero added from unions of cyclotomic classes of order 𝑵𝑵 = 𝟏𝟏𝟏𝟏.

3-adic complexity of ternary generalized cyclotomic sequences with period

In this paper, we study the ternary generalized cyclotomic sequences with a period equal to a power of an odd prime. Ding-Helleseth’s generalized cyclotomic classes of order three are used for the definition of these sequences. We derive the symmetric 3-adic complexity of above mention sequences and obtain the estimate of symmetric 3-adic complexity of sequences. It is shown that 3-adic complexity of these sequences is large enough to resist the attack of the rational approximation algorithm for feedback with carry shift registers.

Notes about symmetric m-adic complexity of generalized cyclotomic sequences of order two with period pq

In this paper, we consider binary generalized cyclotomic sequences with period pq, where p and q are two distinct odd primes. These sequences derive from generalized cyclotomic classes of order two modulo pq. We investigate the generalized binary cyclotomic sequences as the sequences over the ring of integers modulo m for a positive integer m and study m-adic complexity of sequences. We show that they have high symmetric m-adic complexity. Our results generalize well-known statements about 2-adic complexity of these sequences.

On the constructions of MDS self-dual codes via cyclotomy

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.

Pure Gauss sums and skew Hadamard difference sets

Chowla (1962), McEliece (1974), Evans (1977, 1981) and Aoki (1997, 2004, 2012) studied Gauss sums, some positive integral powers of which are in the field of rational numbers. Such Gauss sums are called pure. In particular, Aoki (2004) gave a necessary and sufficient condition for a Gauss sum to be pure in terms of Dirichlet characters modulo the order of the multiplicative character involved. In this paper, we study pure Gauss sums with odd extension degree f and classify them for f=5,7,9,11,13,17,19,23 based on Aoki's theorem. Furthermore, we characterize a special subclass of pure Gauss sums in view of an application for skew Hadamard difference sets. Based on the characterization, we give a new construction of skew Hadamard difference sets from cyclotomic classes of finite fields.

Efficient generation of quadratic cyclotomic classes for shortest quadratic decompositions of polynomials

Efficient generation of quadratic cyclotomic classes for shortest quadratic decompositions of polynomials

Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields

Abstract In 2006, Hubert, Mauduit and Sárközy extended the notion of binary sequences to n-dimensional binary lattices and introduced the measures of pseudorandomness of binary lattices. In 2011, Gyarmati, Mauduit and Sárközy extended the notions of family complexity, collision and avalanche effect from binary sequences to binary lattices. In this paper, we construct pseudorandom binary lattices by using cyclotomic classes in finite fields and study the pseudorandom measure of order k, family complexity, collision and avalanche effect. Results indicate that such binary lattices are “good,” and their families possess a nice structure in terms of family complexity, collision and avalanche effect.

Frequency-Hopping Sequences With Optimal Average Hamming Correlation and Their Applications in Energy and Spectrum Harvesting Technologies Area

The research of cyclotomy theory can be traced to Gauss and it has been applied to many fields such as cryptography, coding theory, and combinatorics. According to $v$ prime numbers or compound words, the incision on the residue-like ring ${\mathbb Z}_{v}$ can be separated to classic incision or general incision. In this work, a kind of extended generalized cyclotomic classes is introduced. Based on this tangent method, a class of frequency hopping sequence set with the best average Hamming correlation is proposed.