Generalizing binary quadratic residue codes to higher power residues over larger fields

In this paper, we provide a generalization of binary quadratic residue codes to the cases of higher power prime residues over the finite field of the same order. We find generating polynomials for such codes, define a new notion corresponding to the binary concept of an idempotent, and use this to find a lower bound for the codeword weight of the duals of such codes. This in turn leads to a lower bound on the weight of the codewords themselves.