Perfect multiple error-correcting arithmetic codes

Abstract:

An arithmetic code is a subgroup of ${{\mathbf {Z}}_{{r^n} \pm 1}}$, with the arithmetic distance $d(x,y) = {\min _t}x - y \equiv \Sigma _{i = 1}^t{c_i}{r^{n(i)}}\;(\bmod {r^n} \pm 1)$, for $|{c_i}| < r$, $n(i) \geqslant 0$ for $1 \leqslant i \leqslant t$. A perfect e-error-correcting code is one from which all $x \in {{\mathbf {Z}}_{{r^n} \pm 1}}$, 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.