Abstract: Theory of structures of Golomb Costas arrays with 1-gap row and 1-gap column was established with the study of the properties and mathematical models of Golomb Costas arrays and their families which are obtained by the cyclic shift method. By conducting a number of researches on the algebraic constructions and the auto- and cross-correlation properties of Golomb Costas arrays with 1-gap row and 1-gap column, certain theorems were confirmed. The design of frequency hopping codes and the distribution of them to the users were illustrated while the design of frequency hopping patterns was explored in OFDM system with Golomb Costas arrays with 1-gap row and 1-gap column. By designing frequency hopping codes with Golomb Costas ayrays with 1-gap row and 1-gap column, an algebraically constructed frequency hop code family can achieve ideal auto-correlation properties. And excellent cross-correlation performance can be obtained when the Doppler shifts are restricted in a wireless communication system.