Algebraic symmetries of generic (m+1)(m+1) dimensional periodic Costas arrays

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

Multidimensional Costas Arrays and Their Enumeration Using GPUs and FPGAs

The enumeration of two-dimensional Costas arrays is a problem with factorial time complexity and has been solved for sizes up to 29 using computer clusters. Costas arrays of higher dimensionality have recently been proposed and their properties are beginning to be understood. This paper presents, to the best of our knowledge, the first proposed implementations for enumerating these multidimensional arrays in GPUs and FPGAs, as well as the first discussion of techniques to prune the search space and reduce enumeration run time. Both GPU and FPGA implementations rely on Costas array symmetries to reduce the search space and perform concurrent explorations over the remaining candidate solutions. The fine grained parallelism utilized to evaluate and progress the exploration, coupled with the additional concurrency provided by the multiple instanced cores, allowed the FPGA (XC5VLX330-2) implementation to achieve speedups of up to 30× over the GPU (GeForce GTX 580)

Multidimensional Costas Arrays and Their Enumeration Using GPUs and FPGAs

The enumeration of two-dimensional Costas arrays is a problem with factorial time complexity and has been solved for sizes up to 29 using computer clusters. Costas arrays of higher dimensionality have recently been proposed and their properties are beginning to be understood. This paper presents, to the best of our knowledge, the first proposed implementations for enumerating these multidimensional arrays in GPUs and FPGAs, as well as the first discussion of techniques to prune the search space and reduce enumeration run time. Both GPU and FPGA implementations rely on Costas array symmetries to reduce the search space and perform concurrent explorations over the remaining candidate solutions. The fine grained parallelism utilized to evaluate and progress the exploration, coupled with the additional concurrency provided by the multiple instanced cores, allowed the FPGA (XC5VLX330-2) implementation to achieve speedups of up to 30× over the GPU (GeForce GTX 580)