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Correlation measure of order $k$ is an important measure of randomness in binary sequences. This measure tries to look for dependence between several shifted version of a sequence. We study the relation between the correlation measure of order $k$ and another two pseudorandom measures: the $N$th linear complexity and the $N$th maximum order complexity. We simplify and improve several state-of-the-art lower bounds for these two measures using the Hamming bound as well as weaker bounds derived from it.

In this paper, we define the linear complexity for multidimensional sequences over finite fields, generalizing the one-dimensional case. We give some lower and upper bounds, valid with large probability, for the linear complexity and $k$-error linear complexity of multidimensional periodic sequences., Comment: 13 pages

We generalize the array orthogonality property for perfect autocorrelation sequences to $n$-dimensional arrays. The generalized array orthogonality property is used to derive a number of $n$-dimensional perfect array constructions.

Single sequences like Legendre have high linear complexity. Known CDMA families of sequences all have low complexities. We present a new method of constructing CDMA sequence sets with the complexity of the Legendre from new frequency hop patterns, and compare them with known sequences. These are the first families whose normalized linear complexities do not asymptote to 0, verified for lengths up to 6x108. The new constructions in array format are also useful in watermarking images. We present a conjecture regarding the recursion polynomials. We also have a method to reverse the process, and from small Kasami/No-Kumar sequences we obtain a new family of 2n doubly periodic (2n+1)x(2n-1) frequency hop patterns with correlation 2., Comment: 10 pages, 8 figures

We introduce a construction for perfect periodic autocorrelation sequences over roots of unity. The sequences share similarities to the perfect periodic sequence constructions of Liu, Frank, and Milewski.

We introduce a construction for periodic zero correlation zone (ZCZ) sequences over roots of unity. The sequences share similarities to the perfect periodic sequence constructions of Liu, Frank, and Milewski. The sequences have two non-zero off-peak autocorrelation values which asymptotically approach $\pm 2 \pi$, so the sequences are asymptotically perfect.

In this work we present two generators for the group of symmetries of the generic $(m+1)$ dimensional periodic Costas arrays over elementary abelian $(\mathbb{Z}_p)^m$ groups: one that is defined by multiplication on $m$ dimensions and the other by shear (addition) on $m$ dimensions. Through exhaustive search we observe that these two generators characterize the group of symmetries for the examples we were able to compute. Following the results, we conjecture that these generators characterize the group of symmetries of the generic $(m+1)$ dimensional periodic Costas arrays over elementary abelian $(\mathbb{Z}_p)^m$ groups.

This paper describes the synthesis of matrices with good correlation, from cyclic shifts of pseudonoise columns. Optimum matrices result whenever the shift sequence satisfies the constant difference property. Known shift sequences with the constant (or almost constant) difference property are: Quadratic (Polynomial) and Reciprocal Shift modulo prime, Exponential Shift, Legendre Shift, Zech Logarithm Shift, and the shift sequences of some m-arrays. We use these shift sequences to produce arrays for watermarking of digital images. Matrices can also be unfolded into long sequences by diagonal unfolding (with no deterioration in correlation) or row-by-row unfolding, with some degradation in correlation.