| 书目名称 | Mathematical Properties of Sequences and Other Combinatorial Structures | | 编辑 | Jong-Seon No,Hong-Yeop Song,P. Vijay Kumar | | 视频video | | | 丛书名称 | The Springer International Series in Engineering and Computer Science | | 图书封面 |  | | 描述 | .Mathematical Properties of Sequences and Other Combinatorial Structures. is an excellent reference for both professional and academic researchers working in telecommunications, cryptography, signal processing, discrete mathematics, and information theory. ..The work represents a collection of contributions from leading experts in the field. Contributors have individually and collectively dedicated their work as a tribute to the outstanding work of Solomon W. Golomb. ..Mathematical Properties of Sequences and Other Combinatorial Structures. covers the latest advances in the widely used and rapidly developing field of information and communication technology.. | | 出版日期 | Book 2003 | | 关键词 | Overlay; Performance; Standard; communication; information; information theory; integrated circuit; number | | 版次 | 1 | | doi | https://doi.org/10.1007/978-1-4615-0304-0 | | isbn_softcover | 978-1-4613-5013-2 | | isbn_ebook | 978-1-4615-0304-0Series ISSN 0893-3405 | | issn_series | 0893-3405 | | copyright | Springer Science+Business Media New York 2003 |
| 1 |
Front Matter |
|
|
Abstract
|
| 2 |
,Pairs of ,-Sequences with a Six-Valued Crosscorrelation, |
Tor Helleseth |
|
Abstract
Let {..} and {..}be two .-sequences of period .. − 1 with symbols from the finite field .(.). The crosscorrelation function between the two .-sequences is defined by . for τ = 0, 1, ..., .. − 2 where ω is a complex .-th root of unity. We determine the distribution of the values of the crosscorrelation function in the case . = .. − .. + .. where . = 4. and .. ≢2 (mod 3).
|
| 3 |
,Permutation Polynomials, Tuscan-, Arrays and Costas Sequences, |
Wensong Chu |
|
Abstract
In this paper, we talk about permutation polynomials over finite fields and their applications in study of circular Tuscan-. arrays and Costas sequences. In the first part, some general characterizations and special families of permutation polynomials over finite fields are discussed. Then we show how permutation polynomials are used to construct circular Tuscan-. arrays. We also present a unified approach to construct Costas sequences, and explore other possible ways to understand and study Costas sequences.
|
| 4 |
,Periods of Piecewise-Linear Recurrences, |
Basil Gordon |
|
Abstract
Let . be a commutative ring with 1. In this note the term .. is used for a difference equation of the form . where the coefficients .. are in .. If .. = 0, the recurrence is called .; otherwise it is .. A solution (.., ..,…) with terms .. ∈ . is called a .; such a sequence is uniquely determined by the initial values .., ..,…, .... A classical example is the Fibonacci sequence {..} = {1, 1, 2, 3, 5,…} with .. = ... + ...; this has degree 2.
|
| 5 |
,Trace Representation of Hall’s Sextic Residue Sequences of Period , ≡ 7 (mod 8), |
Jeong-Heon Kim,Hong-Yeop Song,Guang Gong |
|
Abstract
We determine the trace function representation of Hall‘s sextic residue sequences of period . ≡ 7 (mod 8). Current status of a conjecture regarding the existence of Hadamard sequences is briefly discussed.
|
| 6 |
,Array Correlation and Sequences with Equal Magnitude Correlation, |
Guang Gong |
|
Abstract
In this paper, we will investigate cross correlation of sequences over ℝ where ℝ is either the integer residue ring ℤ.or the Galios ring .(.., .) where ., . and . are positive integers and . is a prime. First, in terms of the interleaved structure of sequences, we will introduce a new type of cross correlation between two periodic sequences over ℝ with period . = . where . > 1 and . > 1 are positive integers, which will be called an . in this paper. Then, we will construct two families of sequences over ℝ of period . = .. We will show that for each pair of sequences, each taken from different sets, their cross correlation (as a usual definition) is equal to their array cross correlation. Moreover, the magnitude of the sum of each correlation value and the constant 1 is equal to the constant . − 1.
|
| 7 |
,Singly Periodic Costas Arrays are Equivalent to Polygonal Path Vatican Squares, |
Herbert Taylor |
|
Abstract
If . + 1 is a prime number with primitive root α, then the function . ≡ .. (mod . + 1) can be plotted in an . × ∞ matrix of dots and blanks where . takes the values 1 to . as . runs through the integers. This was discovered by Lloyd Welch in the mid 70’s in response to a question raised by John P. Costas in a letter to Solomon W. Golomb.
|
| 8 |
,Survey on Costas Arrays and Their Generalizations, |
Oscar Moreno |
|
Abstract
The major constructions of Costas array s are for lengths . = .−l, ..−2, ..−3 for any prime .. The last construction of Costas arrays dates back almost twenty years and there are many . for which no Costas arrays are known. This paper surveys three new concepts: extended Costas arrays, quasi-Costas arrays, and extended quasi-Costas arrays, that have the same functionality as regular Costas arrays. Several major constructions for these new arrays with . = ., .. − 1 for prime ., are surveyed, and, in this manner, fill many gaps for those . for which no Costas array was known..We also survey Moreno, Golomb and Marie’s construction for multiple target arrays. This construction has auto and cross correlation 2. On the other hand it gives arrays of length .. + 1, .., .. − 1 for any prime ...Finally we present some new results on the number of Costas Arrays. We prove that in the tables of existing Costas arrays there are at least 4 arrays, and, we obtain some new constructions of Costas arrays based on Golomb and Taylor’s .. and .. a constructions.
|
| 9 |
,On ,-Ary Bent Functions Defined on Finite Fields, |
Young-Sik Kim,Ji-Woong Jang,Jong-Seon No,Tor Helleseth |
|
Abstract
It is known that a bent function corresponds to a perfect nonlinear function, which makes it difficult to do the differential cryptanalysis in DES and in many other block ciphers. In this paper, for an odd prime ., quadratic .-ary bent functions defined on finite fields are given from the families of .-ary sequences with optimal correlation property. And quadratic .-ary bent functions, that is, perfect nonlinear functions from the finite field .. to its prime field .. are constructed by using the trace functions.
|
| 10 |
,Distribution of ,-Patterns in the Highest Level of ,-Adic Expansion of Some Linear Recursion Sequen |
Zongduo Dai,Dingfeng Ye,Ping Wang,Genxi Fang |
|
Abstract
In this paper, distribution of .-patterns in the highest level of .adic expansion of some linear recursion sequences of degree . over Galois ring . = .(.., ..) is evaluated. As a consequence, this distribution is shown to be asymptotically uniform with respect to the degree ..
|
| 11 |
,On Edit Distance Attack to Alternating Step Generator, |
Shaoquan Jiang,Guang Gong |
|
Abstract
Edit distance between two binary input strings and one binary output string was introduced to attack alternating step generator by Golic and Menicocci. This attack is successful only if the maximal, average and minimal conditional probability of the zero distance, given a key stream of length ., approach zero exponentially. In this paper, using the run property of binary sequences, we prove the average and minimal conditional probability of zero distance exponentially approach zero with .. We also prove if there exists . such that the maximal conditional probability of zero distance of length . is less than ., then the maximal conditional probability of zero distance will exponentially approach zero.
|
| 12 |
,Polyomino Number Theory (II), |
Uldis Barbans,Andris Cibulis,Gilbert Lee,Andy Liu,Bob Wainwright |
|
Abstract
Polyominoes are connected plane figures formed of joining unit squares edge to edge. We have a monomino, a domino, two trominoes named I and V, five tetrominoes named I, L, N, O and T, respectively, and twelve pentominoes (a registered trademark of Solomon W. Golomb) named F, I, L, N, P, T, U, V, W, X, Y and Z respectively.
|
| 13 |
,Achievement Games for Polyominoes on Archimedean Tessellations, |
Jens-P. Bode,Heiko Harborth |
|
Abstract
For the eight Archimedean tessellations of the plane two players . and . alternatingly color the cells. If . has a strategy to achieve a given polyomino . then . is a winner, if . has a strategy to prevent A from winning then . is a loser. A characterization of polyominoes to be a winner or a loser is proved for one tessellation for all polyominoes and for two tessellations for all but 4 polyominoes each. For three tessellations the set of winners is proved to be finite and for two tessellations we obtain upper bounds for the number of winners for small numbers of cells only. In all eight cases the exact numbers of different polyominoes are determined for small numbers of cells.
|
| 14 |
,What’s New in Polyomino Puzzles and Their Design, |
Stewart Coffin,Jerry Slocum |
|
Abstract
The idea of using sets of polyominoes as practical assembly puzzles can be traced back more than two and a half centuries. ., published in 1749, included a cross puzzle made of three Z-pentominoes and two L-polyominoes. The earliest rectangular puzzle formed from polyominoes that we found, named ., was included in Catel’s Catalogue of 1785 .. It consisted of four L-tetrominoes and four Z-pentominoes and the only solution is shown in Figure 1.
|
| 15 |
,Euler Sums Revisited, |
Tom M. Apostol |
|
Abstract
A large literature exists relating Riemann’s zeta function . and partial sums of the harmonic series, . Much of the research originated from two striking formulas discovered by Euler, ., . and a recursion formula, also due to Euler, which states that for integer . we have . For . and .3 this gives ζ(4) =2/5ζ(2). and ζ(6) =8/35ζ(2).. More generally, it shows that ζ(2) is a rational multiple of ζ(2). These results were rediscovered and extended by Ramanujan [.] and many others [.][.][.][.].
|
| 16 |
,Hardy-Littlewood Constants, |
Keith Conrad |
|
Abstract
Extending a technique introduced in Golomb’s thesis, we look at a Dirichlet series naturally associated to the Hardy-Littlewood conjecture.
|
| 17 |
,On the Parity of a Permutation, |
David Singmaster |
|
Abstract
In teaching elementary group theory, the idea of the parity of a permutation . soon arises. Traditional textbooks usually develop this by considering . or . or the sign of.. The last reduces to knowing the number of inversions: ..
|
| 18 |
,Irreducibles of Tetranomial Type, |
Alfred W. Hales,Donald W. Newhart |
|
Abstract
A binary polynomial .(.) will be said to be of . precisely when (. + l).(.) has just four terms. One motivation for this is the common use of tetranomials of the form (. + l).(.), where .(.) is primitive, for cyclic redundancy-check codes (CRC’s); in addition, such polynomials share a simple pattern, namely that all the zero coefficients occur consecutively. This paper analyzes the distribution and algebraic properties of irreducibles of tetranomial type, with particular attention to comparisons with irreducible trinomials. A crucial part of the analysis is an extension of Swan’s Trinomial Theorem to binary tetranomials.
|
| 19 |
,A Method of Deterministically Generating Block Substitution Tables which Meet a Given Standard of N |
Lothrop Mittenthal |
|
Abstract
There is considerable interest in the cryptographic community in block substitution tables or S-boxes, which are highly nonlinear in some sense. This is particularly important in Feistel type systems of which DES is a prime example. In such systems, the key is used to interact with the clear text data and the substitution tables serve as barriers to limit access to the key by comparing clear text with cipher text data. The primary tools of cryptanalysis against Feistel type systems are differential and linear cryptanalysis. The principal foil against these is nonlinearity as typically measured by .. and .. norms using the Walsh-Fourier transform. These highly nonlinear tables are generally found by searching and their properties are determined empirically by testing rather than relying on underlying mathematical theory. The purpose of this paper is to find a method of deterministically generating block substitution tables of maximal nonlinearity, as measured by some recognized criteria without sacrificing other desirable cryptographic qualities.
|
| 20 |
,Tilings with Generalized Lee Spheres, |
Tuvi Etzion |
|
Abstract
We discuss tilings of ℝ. with bodies which are generalizations of the well known Lee spheres. It is shown that if .=2 then there exists a tiling of ℝ. with any generalized Lee spheres of order .. Tilings of ℝ. with generalized Lee spheres with radius one are also shown. All the tilings we consider are based on lattices.
|
|
|
发表于 2025-3-21 18:39:48
