Perfect multiple error-correcting arithmetic codes

Abstract

An arithmetic code is a subgroup of Z r n ± 1 {{\mathbf {Z}}_{{r^n} \pm 1}} , with the arithmetic distance d ( x , y ) = min t x − y ≡ Σ i = 1 t c i r n ( i ) ( mod r n ± 1 ) 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 |{c_i}| > r , n ( i ) ⩾ 0 n(i) \geqslant 0 for 1 ⩽ i ⩽ t 1 \leqslant i \leqslant t . A perfect e-error-correcting code is one from which all x ∈ Z r n ± 1 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.