Multidimensional arrays with maximal linear complexity

Abstract

Multidimensional arrays have proven to be useful in watermarking, therefore interest in this subject has increased in the previous years along with the number of publications. For one dimensional arrays (sequences), linear complexity is regarded as standard measure of complexity. Although linear complexity of sequences has been widely studied, only recently, we have extended it to the study of multidimensional arrays. In this paper, we show that the concept of multidimensional linear complexity is a powerful tool, by examining the results for selected constructs. We have obtained the linear complexity of logartihmic Moreno-Tirkel arrays and we show that they show high multidimensional linear complexity. Finally, we explicitly provide the minimal generators for quadratic Moreno-Tirkel arrays. The results show that these techniques are effective in finding the multidimensional linear complexity of the constructions, representing only a small fraction of the applicability of multidimensional linear complexity. The study of multidimensional arrays provides new ways to understand sequences and set the basis for forthcoming proof of the three years old conjectures related with CDMA sequences.