The combinatorics of binary arrays

This paper gives an account of the combinatorics of binary arrays, mainly concerning their randomness properties. In many cases the problem reduces to the investigation on difference sets.postprin

A Hardware Oriented Method to Generate and Evaluate Nonlinear Interleaved Sequences with Desired properties

It is well known that the combinatorial structure, algebraic structure and D-transform based method render the nonlinear sequences with good autocorrelation function (ACF) and great linear complexity (LC). However, “all sequences” are not equal even if they are “born” by the same method! In this paper the big inequalities regarding LC of these sequences are shown based on a hardware oriented method (D-transform). In order to get the right sequences some more extensive simulations and trade off are needed. That is why this paper is represented here with above Title. Keywords: cryptography, mobile communications, security, watermarking, D-transfor

Difference Balanced Functions and Their Generalized Difference Sets

Difference balanced functions from $F_{q^n}^*$ to $F_q$ are closely related to combinatorial designs and naturally define $p$-ary sequences with the ideal two-level autocorrelation. In the literature, all existing such functions are associated with the $d$-homogeneous property, and it was conjectured by Gong and Song that difference balanced functions must be $d$-homogeneous. First we characterize difference balanced functions by generalized difference sets with respect to two exceptional subgroups. We then derive several necessary and sufficient conditions for $d$-homogeneous difference balanced functions. In particular, we reveal an unexpected equivalence between the $d$-homogeneous property and multipliers of generalized difference sets. By determining these multipliers, we prove the Gong-Song conjecture for $q$ prime. Furthermore, we show that every difference balanced function must be balanced or an affine shift of a balanced function.Comment: 17 page

Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10

AbstractAll Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10 are constructed and classified up to isomorphism together with related Hadamard matrices of order 64. Affine 2-(64,16,5) designs can be obtained from Hadamard 2-(63,31,15) designs having line spreads by Rahilly’s construction [A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991) 291–303]. The parameter set 2-(64,16,5) is one of two known sets when there exists several nonisomorphic designs with the same parameters and p-rank as the design obtained from the points and subspaces of a given dimension in affine geometry AG(n,pm) (p a prime). It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2-(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3,4), or is one of the two exceptional designs found by Harada, Lam and Tonchev [M. Harada, C. Lam, V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs Codes Cryptogr. 34 (2005) 71–87]

Two-tuple balance of non-binary sequences with ideal two-level autocorrelation

AbstractLet p be a prime, q=pm and Fq be the finite field with q elements. In this paper, we will consider q-ary sequences of period qn-1 for q>2 and study their various balance properties: symbol-balance, difference-balance, and two-tuple-balance properties. The array structure of the sequences is introduced, and various implications between these balance properties and the array structure are proved. Specifically, we prove that if a q-ary sequence of period qn-1 is difference-balanced and has the “cyclic” array structure then it is two-tuple-balanced. We conjecture that a difference-balanced q-ary sequence of period qn-1 must have the cyclic array structure. The conjecture is confirmed with respect to all of the known q-ary sequences which are difference-balanced, in particular, which have the ideal two-level autocorrelation function when q=p

A note on low correlation zone signal sets

Abstract. In this note, we present a connection between designing low correlation zone (LCZ) sequences and the results of correlation of sequences with subfield decompositions presented in a recent book by the first two authors [2]. This results in low correlation zone signal sets with huge sizes over three different alphabetic sets: finite field of size q, integer residue ring modulo q, and the subset in the complex field which consists of powers of a primitive q-th root of unity. We also provide two open problems along this direction. Index Terms: low correlation zone sequences, subfield reducible sequences, two-tuple balance property.