Hadamard matrices are n × n matrices with elements in {1, -1} for which the inverse of the matrix is the transpose scaled by 1/n. The most important conjecture in the theory of Hadamard matrices is their existence when n is a multiple of 4. Although algebraic constructions have been proposed for some specific values, no general constructions exist in the literature, and they are usually found by computational search for a given n. The smallest value for which a Hadamard matrix of order n is not known, is n = 668.

Families of binary sequences with good auto- or cross-correlation properties are required in areas such as cryptography, wireless communications, and digital watermarking. Algebraic constructions present an advantage over random sequences in that fixed bounds can be provided for the off-peak correlation. The interleave and the composition method take a well-distributed pseudonoise sequence, such as an m-sequence, that can be extended to produce larger families with given properties and longer periods. The interleave method or row-by-row folding of a sequence is a popular approach to generate families such as Gordon Mill Welch (GMW) sequences, which are constructed using a shift sequence with good correlation properties. The method of composition (a different approach) also requires a shift sequence and a pseudonoise sequence to construct a new family more related with multidimensional periodic arrays. In this work, we develop new families of binary sequences with good pseudorandom properties based on both the interleave and composition methods, which offer flexible periods and easy implementation. This new construction is based on the generalized Extended Rational Cycle (ERC) construction developed by Rivat and Niederreiter. The results show that for certain parameters the obtained families are close to optimal in terms of family size and correlation.

Costas arrays are fundamental to the operation of radar and sonar systems, yet it is still an open problem how to generate a Costas array for an arbitrary length. A way to improve computer search is to make use of a characterization of the structural properties in order to reduce the search space. In this work, we use techniques from uniform distribution theory to study several structural constraints on Welch and Golomb constructions of Costas arrays. Then, we find bounds on the number of streaks of a given length and their deficiency and test their tightness with several numerical experiments. These results partially answer questions posed by Correll, Jedwab, and Wodlinger.

Families of binary sequences with low correlation are required in applications such as wireless communications, ranging and time delay measurement and digital watermarking, among others. Many constructions have been proposed that employ m-sequences as basic building blocks, such as the Gordon-Mills-Welch (GMW) sequences. In this work we present a unified construction of GMW sequences derived from suitable m-sequences by using the method of composition, producing sequences of length$2^{n}-1$with$n$being an integer composite number. A given m-sequence is folded using the Chinese remainder theorem (CRT) into a two dimensional array, whose columns are either constant or cyclic shifts of a short$m$-sequence. Then, the array can be summarized by a short$m$-sequence and the sequence of shifts (shift sequence). By looking at array interpretation, we avoid using Trace representation of intermediate fields, which makes its implementation more straightforward and secure. Pseudonoise arrays can be produced by constructing all the valid columns and shift sequences, the latter obtained by proper decimations and proper multiplication. Equivalences are removed by selecting a cyclotomic set leader from the degenerate conjugacy class. The window properties of these sequences can be exploited to construct generalized GMW sequence generators for lengths as large as the long codes in GPS i.e.$2^{42}-1$. A similar generalisation of the small Kasami sets can be constructed using this algorithm. The complexity of the algorithms is$\sqrt{L}$where$L$is the length of the sequence, improving known algorithms. Finally, the shift sequences constructed in this work are good candidates to frequency hopping and time hopping sequences in UWB wireless communications and localisation systems.