New results on nonexistence of perfect p-ary sequences and almost p-ary sequences

Complex periodical sequences with lower autocorrelation values are used in CDMA communication systems and cryptography. In this paper we present new nonexistence results on perfect p-ary sequences and almost p-ary sequences and related difference sets by using some knowledge on cyclotomic fields and their subfields.     

本刊英文版Vol.32(2016),No.1论文摘要（英文）

正New Results on Nonexistence of Perfect p-Ary Sequences and Almost p-Ary Sequences Hal Ying LIU Ke Qin FENG Abstract Complex periodical sequences with lower autocorrelation values are used in CDMA communication systems and cryptography.In this paper we present new nonexistence results on perfect p-ary sequences and almost p-ary sequences and related difference sets by using some knowledge on cyclotomic fields and their subfields.     

On equivalence of negaperiodic Golay pairs

Associated pairs as defined by Ito (J Algebra 234:651–663, 2000) are pairs of binary sequence of length 2t satisfying certain autocorrelation properties that may be used to construct Hadamard matrices of order 4t. More recently, Balonin and Dokovi? (Inf Control Syst 5:2–17, 2015) use the term negaperiodic Golay pairs. We define extended negaperiodic Golay pairs and prove a one-to-one correspondence with central relative (4t, 2, 4t, 2t)-difference sets in dicyclic groups of order 8t. We present a new approach for computing negaperiodic Golay pairs up to equivalence, and determine conditions where equivalent pairs correspond to equivalent Hadamard matrices. We complete an enumeration of negaperiodic Golay pairs of length 2t for \(1 \le t \le 10\), and sort them into equivalence classes. Some structural properties of negaperiodic Golay pairs are derived.     

广义差集与几乎完美序列

提出了广义差集的概念,并且给出了广义差集的一些初等性质.从应用的角度讲,广义差集就是使得其±1特征序列的自相关函数是(最多)三值的一种组合结构.因此,广义差集不仅仅是在概念(理论)上的推广,它还具有深层次的应用背景.事实上,给出了一些广义差集,它不是可分差集,也不是相对差集.同时也给出了一类广义差集存在的一些必要条件,使得这些广义差集对应的±1特征序列成为几乎完美序列.并举例说明本文中的方法是有效的.     

Optimal and perfect difference systems of sets from <Emphasis Type="Italic">q</Emphasis>-ary sequences with difference-balanced property

Difference systems of sets (DSS) are important for the construction of codes for synchronization. In this paper, a general construction of optimal and perfect difference systems of sets based on q-ary sequences of period n = −1 (mod q) with difference- balanced property is presented, where q is a prime power. This works for all the known q-ary sequences with ideal autocorrelation, and generalizes the earlier construction based on ternary sequences with ideal autocorrelation. In addition, we construct another class of optimal and perfect difference systems of sets, employing decimation of q-ary d-form sequences of period q ^m −1 with difference-balanced property, which generalizes the previous construction from power functions.     

非线性微分方程正解的不存在性

本文研究了非线性微分方程正解不存在性     

Existence status of some previously open abelian difference sets

We theoretically establish the existence status of some previously open abelian difference sets. More precisely, we show the nonexistence of all the following difference sets: (841,120,17) in Z₂₉×Z₂₉, (364,121,40) in Z₂×Z₂×Z₇×Z₁₃, (837,133,21) in Z₃×Z₃×Z₃×Z₃₁,Z₃×Z₉×Z₃₁ and Z₈₃₇.     

Answering a Question of Pott on Almost Perfect Sequences

Periodic binary (plus-minus) sequences all but one of whose out-of-phase autocorrelation coefficients are zero are studied by Wolfman [6]. Using the equivalence of these almost perfect sequences to certain cyclic divisible difference sets (noted by Bradley and Pott [1]), we settle the existence status of a perviously open case of an almost perfect sequence of length 852, thereby answering a question of Pott [5] negatively.     

Ⅱ型循环拟差集

本文得到了Ⅱ型循环拟差集的一个基本性质和两个重要的不存在性定理。     

Exhaustive construction of (255, 127, 63)-cyclic difference sets

Here, (255, 127, 63)-cyclic difference sets are exhaustively constructed. There are, in total, 64 distinct (255, 127, 63)-cyclic difference sets. They include one Singer's type and three miscellaneous types. These cyclic difference sets of miscellaneous types also provide new examples of Hadamard matrixs of order 256 and two-level autocorrelation sequences of length 255.     

Constructions of Partial Difference Sets and Relative DifferenceSets Using Galois Rings

We use Galois rings to construct partial difference sets and relative difference sets in non-elementary abelianp-groups. As an example, we also use Galois ringG R(4, 2) to construct a (96,20,4) difference set in Z₄ × Z₄ × Z₆.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

New type of block design

We are already familiar with (υ, k, λ)‐difference sets and (υ, k, λ)‐designs. In this paper, we will introduce a new class of difference sets and designs: (υ, k, [λ₁, λ₂, … , λ_m])‐difference sets and (υ, k, [λ₁,λ₂, … , λ_m])‐designs. We will mainly study designs with a relationship we call λ‐equivalence and use them to produce other designs. Some existence or nonexistence theorems will be given. © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 1–23, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10031     

Exhaustive determination of (1023, 511, 255)-cyclic difference sets

An exhaustive search for (1023, 511, 255)-cyclic difference sets has been conducted. A total of 10 non-equivalent (1023, 511, 255)-cyclic difference sets have been found, all of which are members of previously known or conjectured infinite families. A fast and effective autocorrelation test method was utilized that can also facilitate the testing of longer sequences.

     

Construction of low-hit-zone frequency hopping sequences with optimal partial Hamming correlation by interleaving techniques

In quasi-synchronous frequency hopping code division multiple-access systems, frequency hopping sequences (FHSs) with low-hit-zone (LHZ) are commonly employed to minimize multiple-access interferences. Usually, the length of correlation window is shorter than the period of the chosen FHSs due to the limited synchronization time or hardware complexity. Thus the study of the partial Hamming correlation properties of LHZ FHSs is of particular importance. In this paper, we prove the nonexistence of LHZ FHS sets with strictly optimal partial Hamming correlation in some conditions. In addition, employing the interleaving techniques, we present a construction of LHZ FHS sets with optimal partial Hamming correlation from m-sequences. The fundamental idea of our design is to use short individual m-sequences together with certain appropriate shift sequences to construct long LHZ FHS sets with optimal partial Hamming correlation. In particular, our construction gives new parameters not covered in the literature.     

McFarland's conjecture on abelian difference sets with multiplier −1

This paper is a continuation of the work by R.L. McFarland and S.L. Ma on abelian difference sets with –1 as a multiplier. More nonexistence results are obtained as a consequence of a theorem on the existence of sub-difference sets. In particular, nonexistence is shown for the two cases left undecided by McFarland and Ma.     

Regular groups on generalized quadrangles and nonabelian difference sets with multiplier -1

This is a first approach to the study of regular generalized quadrangles (i.e. generalized quadrangles with an automorphism group sharply 1-transitive on points). In this paper we point out how the problem is connected to the theory of difference sets with multiplier-1. First, some of the results in [3] on difference sets with multiplier-1 are extended to the nonabelian case; then, these new results on difference sets are used to prove nonexistence theorems for regular GQs of even order s=t.Dedicated to Otto Wagner on the occasion of his 60th birthday     

Statistical Weakness of Multiplexed Sequences

It is proved that the out-of-phase autocorrelation of a well-known class of multiplexed sequences for cryptographic and spread spectrum applications is necessarily high for relatively small values of the phase shifts. Related design criteria for the multiplexer generator including the use of full positive difference sets are proposed. By generalizing the classical occupancy problem, higher order statistical weaknesses of multiplexed sequences are also established.     

Periodic solutions of a 2nth-order nonlinear difference equation

In this paper, a 2 nth-order nonlinear difference equation is considered.Using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of periodic solutions.Results obtained complement or improve the existing ones.     

Semicontinuity of a functional depending on curvature

In this paper we prove a lower semicontinuity result for a functional , defined on a class of bounded subsets of with a piecewise boundary, with respect to the -convergence of the sets. The functional depends on the curvature of in a linear way and contains a penalizing term which prevents the appearance of thin sets in the symmetric difference , where in an -approximating sequence of . Received: 3 January 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001     

Nonexistence and existence of solutions for a fourth-order discrete mixed boundary value problem

In this paper, a fourth-order nonlinear difference equation is considered. By using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of solutions for mixed boundary value problem and give some new results. Results obtained generalize and complement the existing ones.