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An arithmetic code is a subgroup of Zr" + l, with the arithmetic distance d(x, y) = min. X Y c, r() (mod r' + 1), for Icl 0 for 1 < i < t. A perfect eerror-correcting code is one from which all x E Zr ? + l are within distance e of exactly one codeword. Necessary and sufficient (assuming the Generalized Riemann Hypothesis) conditions for the existence of infinitely many perfect single error-correcting codes for a given r are known. In this paper some conditions for the existence of perfect multiple error-correcting codes are given, as well as the results of a computer search for examples.

Perfect multiple error-correcting arithmetic codes