Algebraic Symmetries of Generic $(m+1)$-Dimensional Periodic Costas Arrays

Abstract

In this work we present two generators for the group of symmetries of the generic (m + 1) dimensional periodic Costas arrays over elementary abelian (Zp) 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 (Zp) m groups. and rafael.arce@upr.edu.; O. Moreno is with the Gauss

Multidimensional Costas Arrays and Their Enumeration Using GPUs and FPGAs

“…All enumerated MPCAs turned out to be either Welch Costas constructions as presented in [13] or their symmetries introduced in [7], that is, no spurious MPCAs were found similar to some sizes of 2DCAs. These results support the conjecture that MPCAs (of all sizes and dimensions) are fully characterized by multidimensional Welch Costas arrays along with their symmetries.…”

“…With each new size enumerated, new properties and generation techniques may be discovered [6]. Ortiz-Ubarri et al presented MPCA transformations and their first enumeration in [7]. Given their relatively new discovery, it is expected 2 International Journal of Reconfigurable Computing that the enumeration of MPCAs will, as with 2DCAs, help researchers improve their understanding.…”

“…Both 2DCAs and MPCAs can be generated using algebraic constructions based on finite fields like the Welch and Lempel-Golomb constructions [7]. Small sizes can be enumerated using hand computation, yet the only known method to guarantee complete enumeration is by exhaustive exploration.…”

Multidimensional Costas Arrays and Their Periodicity

“…, a m ). The order of a permutation array A, denoted n, is the number of dots in A, that is, [6,15,18,24] lack at least one of the aforementioned features. A downside of our definition is that we do not know any systematic way of constructing multidimensional Costas arrays other than the reshaping technique described in [6, §4] for the special case of arrays with m even and…”

“…To obtain a higher-dimensional analog of Costas arrays one has to generalize the two defining properties: being a permutation array and having no repeated difference vectors, i.e., the Costas condition. Some multidimensional analogs of Costas arrays have been proposed before [1,6,15,18,24], all satisfying the same multidimensional Costas condition, as it generalizes naturally; however, they differ in the generalization of a permutation array, as this can be done in different ways. Nonetheless, the generalization in [15, §2], which produces arrays of the type defined in [7, Definition 8], have an extremely low density of 1's, thus these arrays "tend not to be very interesting" [7, p. 4].…”