5
2014.
On the characterization of a semi-multiplicative analogue of planar functions over finite fields
In this paper, we present a characterization of a semi-multiplicative analogue of planar functions over finite fields. When the field is a prime field, these functions are equivalent to a variant of a doubly-periodic Costas array and so we call these functions Costas. We prove an equivalent conjecture of Golomb and Moreno that any Costas polynomial over a prime field is a monomial. Moreover, we give a class of Costas polynomials over extension fields and conjecture that this class represents all Costas polynomials. This conjecture is equivalent to the conjecture that there are no non-Desarguesian planes of a given type with prime power order.