A square root bound on the minimum weight in quasi-cyclic codes

We establish a square root bound on the minimum weight in the quasi-cyclic binary codes constructed by Bhargava, Tavares, and Shiva. The proof rests on viewing the codes as ideals in a group algebra over GF (4). Theorem 6 answers a question raised by F. J. MacWilliams and N. J. A. Sloane in {\em The Theory of Error-Correcting Codes.} Theorems 3, 4, and 5 provide information about the way the nonzero entries of a codeword of minimum weight are distributed among the coordinate positions.