A $[2r,2]$ linear code over $\operatorname{GF}(q)$ is called double circulant if it can be generated by $[I,A]$ where $A$ is an $r\times r$ circulant matrix. In this paper, a Fourier transform technique is employed to construct double circulant codes and a lower bound on the minimum distance of such codes is obtained. Further, a decoding process of $[n,n-r]$ RS codes over $\operatorname{GF}(q)$ is described that is based on a representation of parity-check by circulant blocks, using the encoding circuit and Blahut's decoder.