Let T be a complete local ring and C a finite set of incomparable prime ideals of T . We find necessary and sufficient conditions for T to be the completion of an integral domain whose generic formal fiber is semilocal with maximal ideals the elements of C. In addition, if the characteristic of T is zero, we give necessary and sufficient conditions for T to be the completion of an excellent integral domain whose generic formal fiber is semilocal with maximal ideals the elements of C.

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.

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