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

2021 ◽  
Vol 2052 (1) ◽  
pp. 012007
Author(s):  
V A Edemskiy ◽  
S V Garbar
Keyword(s):  
Positive Integer ◽  
Ring Of Integers ◽  
Generalized Cyclotomic Classes ◽  
Cyclotomic Sequences

Abstract 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.

A new class of quaternary generalized cyclotomic sequences of order 2d and length 2pm with high linear complexity

2018 ◽  
Vol 16 (01) ◽  
pp. 1850006 ◽  
Author(s):  
Longfei Liu ◽  
Xiaoyuan Yang ◽  
Bin Wei ◽  
Liqiang Wu
Keyword(s):  
Finite Fields ◽  
Linear Complexity ◽  
Periodic Sequences ◽  
New Class ◽  
New Family ◽  
Generalized Cyclotomic Classes ◽  
Cyclotomic Sequences

Periodic sequences over finite fields, constructed by classical cyclotomic classes and generalized cyclotomic classes, have good pseudo-random properties. The linear complexity of a period sequence plays a fundamental role in the randomness of sequences. In this paper, we construct a new family of quaternary generalized cyclotomic sequences with order [Formula: see text] and length [Formula: see text], which generalize the sequences constructed by Ke et al. in 2012. In addition, we determine its linear complexity using cyclotomic theory. The conclusions reveal that these sequences have high linear complexity, which means they can resist linear attacks.

scholarly journals 3-adic complexity of ternary generalized cyclotomic sequences with period

2021 ◽  
Vol 2052 (1) ◽  
pp. 012008
Author(s):  
V A Edemskiy ◽  
S A Koltsova
Keyword(s):  
Approximation Algorithm ◽  
Rational Approximation ◽  
Generalized Cyclotomic Classes ◽  
Definition Of ◽  
Cyclotomic Sequences

Abstract 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.

scholarly journals Autocorrelation Values of Generalized Cyclotomic Sequences with Period pn+1

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 950
Author(s):  
Xiaolin Chen ◽  
Huaning Liu
Keyword(s):  
Positive Integer ◽  
Linear Complexity ◽  
Character Sums ◽  
Binary Sequences ◽  
Square Sequences ◽  
Cyclotomic Sequences

Recently Edemskiy proposed a method for computing the linear complexity of generalized cyclotomic binary sequences of period p n + 1 , where p = d R + 1 is an odd prime, d , R are two non-negative integers, and n > 0 is a positive integer. In this paper we determine the exact values of autocorrelation of these sequences of period p n + 1 ( n ≥ 0 ) with special subsets. The method is based on certain identities involving character sums. Our results on the autocorrelation values include those of Legendre sequences, prime-square sequences, and prime cube sequences.

On étale covers of curves

1999 ◽  
Vol 127 (1) ◽  
pp. 1-5
Author(s):  
S. KAMIENNY ◽  
J. L. WETHERELL
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Abelian Varieties ◽  
Ring Of Integers ◽  
The Relationship

Let K be a number field with ring of integers R. For each integer g>1 we consider the collection of abelian, étale R-coverings f[ratio ]Y→X, where X and Y are connected proper curves over R and the genus of X is g. We ask the following question: is there a positive integer B = B(K, g) which bounds the degree of such coverings? In this note we provide partial results towards such a bound and study the relationship with bounds on torsion in abelian varieties.

Free quotients of subgroups of the Bianchi groups whose kernels contain many elementary matrices

1994 ◽  
Vol 116 (2) ◽  
pp. 253-273
Author(s):  
A. W. Mason ◽  
R. W. K. Odoni
Keyword(s):  
Normal Subgroup ◽  
Positive Integer ◽  
Commutator Subgroup ◽  
Quadratic Number ◽  
Congruence Subgroups ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
Imaginary Quadratic Number ◽  
Low Dimensional ◽  
Fundamental Paper

AbstractLet d be a square-free positive integer and let be the ring of integers of the imaginary quadratic number field ℚ(√ − d) The Bianchi groups are the groups SL2() (or PSL2(). Let m be the order of index m in . In this paper we prove that for each d there exist infinitely many m for which SL2(m)/NE2(m) has a free, non-cyclic quotient, where NE2(m) is the normal subgroup of SL2(m) generated by the elementary matrices. When d is not a prime congruent to 3 (mod 4) this result is true for all but finitely many m. The proofs are based on the fundamental paper of Zimmert and its generalization due to Grunewald and Schwermer.The results are used to extend earlier work of Lubotzky on non-congruence subgroups of SL2(), which involves the concept of the ‘non-congruence crack’. In addition the results highlight a number of low-dimensional anomalies. For example, it is known that [SLn(m), SLnm)] = En(m), when n ≥ 3, where [SLn(m), SLn(m)] is the commutator subgroup of SL(m) and En(m) is the subgroup of SLn(m) generated by the elementary matrices. Our results show that this is not always true when n = 2.

Automorphisms of the zero-divisor graph of 2 × 2 matrix ring over ℤps

2017 ◽  
Vol 16 (12) ◽  
pp. 1750227 ◽  
Author(s):  
Hengbin Zhang ◽  
Jizhu Nan ◽  
Gaohua Tang
Keyword(s):  
Positive Integer ◽  
Directed Graph ◽  
Zero Divisor ◽  
Matrix Ring ◽  
Directed Edge ◽  
Ring Of Integers ◽  
Zero Divisor Graph ◽  
Zero Divisors

Let [Formula: see text] be the ring of integers modulo [Formula: see text] where [Formula: see text] is a prime and [Formula: see text] is a positive integer, [Formula: see text] the [Formula: see text] matrix ring over [Formula: see text]. The zero-divisor graph of [Formula: see text], written as [Formula: see text], is a directed graph whose vertices are nonzero zero-divisors of [Formula: see text], and there is a directed edge from a vertex [Formula: see text] to a vertex [Formula: see text] if and only if [Formula: see text]. In this paper, we completely determine the automorphisms of [Formula: see text].

2-Adic and Linear Complexities of a Class of Whiteman’s Generalized Cyclotomic Sequences of Order Four

2019 ◽  
Vol 30 (05) ◽  
pp. 759-779
Author(s):  
Priti Kumari ◽  
Pramod Kumar Kewat
Keyword(s):  
Linear Complexity ◽  
Binary Sequences ◽  
Random Sequences ◽  
Generalized Cyclotomic Classes ◽  
Order Four ◽  
Cyclotomic Sequences

Although for more than 20 years, Whiteman’s generalized cyclotomic sequences have been thought of as the most important pseudo-random sequences, but, there are only a few papers in which their 2-adic complexities have been discussed. In this paper, we construct a class of binary sequences of order four with odd length (product of two distinct odd primes) from Whiteman’s generalized cyclotomic classes. After that, we determine both 2-adic complexity and linear complexity of these sequences. Our results show that these complexities are greater than half of the period of the sequences, therefore, it may be good pseudo-random sequences.

On a problem of MacDowell and Specker

1980 ◽  
Vol 45 (3) ◽  
pp. 612-622 ◽  
Author(s):  
Mark Nadel
Keyword(s):  
Positive Integer ◽  
Standard Model ◽  
Complete Characterization ◽  
The Other ◽  
Similar Question ◽  
Ring Of Integers ◽  
Positive Elements ◽  
The Standard Model ◽  
Distinguished Element ◽  
Ordered Ring

Let T extend the theory P of Peano arithmetic, and suppose . Form from a model , in analogy to the way in which the ordered ring of integers is formed from the standard model of arithmetic. Let P′ and T′ be the corresponding analogues of P and T respectively. Now consider the group . In [5] MacDowell and Specker set out to determine the structure of such groups. (The precise statement in [5] refers to the ring of integers rather than the ordered ring. However, as pointed out to us by J. Knight, since Lagrange's Theorem that a positive integer is the sum of four squares is provable in the analogue of P′ for rings (see, for example, the proof in [7, p. 102]), the set of positive elements is definable in the ring, and consequently, so is the ordering. Thus, for the present purpose it makes no difference which of the two structures is used. Of course, one needs the ordering to discuss end extensions, as considered in [5]. On the other hand, one should be aware that in Pr′ one cannot define an ordering, where the theory Pr′ is the theory of the group of integers with distinguished element 1, 〈Z, +, 1〉. The constant 1 is needed so that divisibility mod n can be expressed. We will return to this point later.) In §1 we shall outline the results in this direction obtained in [5].Lipshitz and Nadel, unaware that a similar question had been posed and investigated in [5] (though, of course aware that [5] contained the celebrated results on end extensions) set out to characterize those models 〈A, +〉 of Pr = Presburger Arithmetic (the complete theory of 〈ω, +〉) which can be expanded to models 〈A, +, ·, 0, 1, ≤〉 of P. They were able to give a complete characterization for countable models 〈A, +〉 in [4], which we describe in §2.

PRIMITIVE RECURSIVE DECIDABILITY FOR THE RING OF INTEGERS OF THE COMPOSITUM OF ALL SYMMETRIC EXTENSIONS OF ℚ

2020 ◽  
pp. 1-6
Author(s):  
MOSHE JARDEN ◽  
AHARON RAZON
Keyword(s):  
Positive Integer ◽  
Galois Group ◽  
Galois Extensions ◽  
Ring Of Integers ◽  
Primitive Recursive

Abstract Let ℚsymm be the compositum of all symmetric extensions of ℚ, i.e., the finite Galois extensions with Galois group isomorphic to S n for some positive integer n, and let ℤsymm be the ring of integers inside ℚsymm. Then, TH(ℤsymm) is primitive recursively decidable.

scholarly journals Units and Cyclotomic Units in Zp-Extensions

1995 ◽  
Vol 140 ◽  
pp. 101-116 ◽  
Author(s):  
Jae Moon Kim
Keyword(s):  
Positive Integer ◽  
Galois Group ◽  
Index Theorem ◽  
Unit Group ◽  
Cohomology Groups ◽  
Ring Of Integers ◽  
Basic Properties ◽  
Cyclotomic Units

Let p be an odd prime and d be a positive integer prime to p such that d ≢ 2 mod 4. For technical reasons, we also assume that . For each integer n ≥ 1, we choose a primitive nth root ζn of 1 so that whenever n | m. Let be its cyclotomic Zp-extension, where is the nth layer of this extension. For n ≤ 1, we denote the Galois group Ga\(Kn/K0) by Gn, the unit group of the ring of integers of Kn by En, and the group of cyclotomic units of Kn by Cn. For the definition and basic properties of cyclotomic units such as the index theorem, we refer [6] and [7]. In this paper we examine the injectivity of the homomorphism between the first cohomology groups induced by the inclusion Cn → En.

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