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The binary linear feedback shift register sequences of degree and maximum period  = 2 − 1 (the ) are useful in numerous applications because, although deterministic, they satisfy a number of interesting “randomness properties”.

An important open question is whether a binary sequence of period  = 2 − 1 with both the span- property and the two-level correlation property must be an -sequence.

There is a direct correspondence between -sequences of degree and primitive polynomials of degree over (2). Several conjectures are presented about primitive polynomials with a bounded number of terms.

Bent functions achieve largest distance to all linear functions. Equivalently, their spectrum with respect to the Hadamard-Walsh transform is flat (i.e. all spectral values have the same absolute value). That is equivalent to saying that the function has optimum periodic autocorrelation properties. Negaperiodic correlation properties of are related to another unitary transform called the nega-Hadamard transform. A function is called if the spectrum under the nega-Hadamard transform is flat. In this paper, we consider functions which are simultaneously bent and negabent, i.e. which have optimum periodic and negaperiodic properties. Several constructions and classifications are presented.

Let be a positive integer. A nonzero element of the finite field of order  = 2 is said to be “strongly primitive” if every element ( + )/( + ), with ,,, in {0,1} and  −  not zero, is primitive in the usual sense. We show that the number of such strongly primitive elements is asymptotic to ′· where is the product of (1 − 1/) over all primes dividing ( − 1) and ′ is the product of (1 − 2/) over the same set.

Using this result and the accompanying error estimates, with some computer assistance for small , we deduce the existence of such strongly primitive elements for all except  = 1, 4, 6. This extends earlier work on Golomb’s conjecture concerning the simultaneous primitivity of and  + 1.

We also discuss analogous questions concerning strong primitivity for other finite fields.

The perfect binary sequence of period 4 − ‘0111’ (or cyclic shifts of itself or its complement) − has the optimal periodic autocorrelation function where all out-of-phase values are zero. Not surprisingly, it is also the Barker sequence of length 4 where all out-of-phase aperiodic autocorrelation values have the magnitudes of at most one. From these observations, the applications of the sequence for low periodic and aperiodic autocorrelations are studied. First, the perfect sequence is discussed for binary sequences with optimal periodic autocorrelation. New binary sequences of period  = 4(2 − 1),  = 2 with optimal periodic autocorrelation are presented, which are obtained by a slight modification of product sequences of binary -sequences and the perfect sequence. Then, it is observed that a product sequence of the Legendre and the perfect sequences has not only the optimal periodic but also the good aperiodic autocorrelations with the asymptotic merit factor 6. Moreover, if the product sequences replace Legendre sequences in Borwein, Choi, Jedwab (BCJ) sequences, or equivalently Kristiansen-Parker sequences (simply BCJ-KP sequences), numerical results show that the resulting sequences have the same asymptotic merit factor as the BCJ-KP sequences.

Considered are quadratic -ary bent functions having the form . Described is the general Gold-like class of bent functions that covers all the previously known monomial quadratic cases. Obtained is the exact value of the Walsh transform coefficients for a bent function in this class. In particular, presented is an explicit expressions for a dual of a monomial quadratic bent function which is a bent functions on its own. This gives new examples of generalized bent functions not previously reported in the literature. The paper is the follow-up to Helleseth-Kholosha 2006.

In this paper, for a positive odd integer , a new family of binary sequences with 2 + 1 sequences of length 2 − 1 taking three level nontrivial correlations − 1 and − 1±2 is presented, whose correlation distribution is the same as that of the well-known Gold sequences. This family may be considered as a new class of Gold-like sequences.

A motivation for this paper is to find a practical mechanism for generating permutations of a finite set of consecutive positive integers so that the resultant spacings between originally consecutive numbers and  + , are now different for each . This is equivalent to finding complete Latin squares. The ordered set of spacings in such a permutation is called a sequencing. The set of partial sums of the terms in a sequencing is called a directed terrace. For lack of standard terminology, the associated permutations here are called quick trickles.

This paper concerns methods of finding such sequencings, in part by finding constraints on their existence, so that search time can be substantially reduced. A second approach is to represent a quick trickle permutation as a directed graph. The sequencings then are represented by chords of different lengths. Various methods can be used to rearrange the chords and obtain additional sequencings and groups of quick trickle permutations.

It is well known that complete Latin squares of size × can be found if is even but not if is odd. This is equivalent to saying that the group of integers (1, 2, 3,..., ) under the operation of multiplication mod is sequenceable if and only if is even. A more general concept is introduced which is termed quasi-sequenceable. This applies to both even and odd sizes.

The application to block encryption, or the so-called substitution permutation system is briefly described. The net result is interround mixing, deterministically generated under key control, and quickly replaceable with a new pattern of equal merit.

Currently, typical block encryption systems use algorithms which are fixed and publicly known. The motivation here is to develop block encryption systems using algorithms which are variable, secret, generated and periodically changed by the key.

Digital watermarking applications require constructions of double-periodic matrices with good correlations. More specifically we need as many matrix sequences as possible with both good auto- and cross-correlation. Furthermore it is necessary to have double-periodic sequences with as many dots as possible.

We have written this paper with the specific intention of providing a theoretical framework for constructions for digital watermarking applications.

In this paper we present a method that increases the number of sequences, and another that increases the number of ones keeping the correlation good and double-periodic. Finally we combine both methods producing families of double-periodic arrays with good correlation and many dots. The method of increasing the number of sequences is due to Moreno, Omrani and Maric. The method to increase the number of dots was started by Nguyen, Lázló and Massey, developed by Moreno, Zhang, Kumar and Zinoviev, and further developed by Tirkel and Hall. The very nice application to digital watermarking is due to Tirkel and Hall.

Finally we obtain two new constructions of Optical Orthogonal Codes: Construction A which produces codes with parameters and Construction B which produces families of code with parameters and family size  + 1.

In phase-encoded optical CDMA (OCDMA) spreading is achieved by encoding the phase of signal spectrum. Here, a mathematical model for the output signal of a phase-encoded OCDMA system is first derived. This is shown to lead to a performance metric for the design of spreading sequences for asynchronous transmission.

Generalized bent functions are used to construct a family of efficient phase-encoding sequences. It is shown how M-ary modulation of these spreading sequences is possible. The problem of designing efficient phase-encoded sequences is then related to the problem of minimizing PMEPR (peak-to-mean envelope power ratio) in an OFDM communication system.

() is the maximum number of patterns that can sit on the × board where each pattern consists of nonattacking Queens placed symmetrically around the center. Each square of the board has at most one Queen. () is the same except that “placed symmetrically around the center” is not required.