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

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 \(m < 2^{33} – 1\).