On Perfect Binary Arithmetic Codes Which Can Correct Two Errors or More

Abstract

We state here that, for modulus $m$ odd and less than $2^{29}+2^{27}–1$, no (nontrivial) perfect binary arithmetic code, correcting two errors or more, exists (this is to be taken with respect to the Garcia-Rao modular distance). In particular, in the case $m=2^n\pm 1$, which is most frequently studied, no such code exists for $m<2^{33}–1$.