On Varshamov–Tenengolts Codes and a Conjecture of L. A. Bassalygo

We consider a construction problem for binary codes that correct single localized errors. L. A. Bassalygo stated a conjecture that the maximum “cardinality” (the number of messages) of such a code is equal to the integral part of the corresponding value of the Hamming bound. Using Varshamov–Tenengolts codes, we prove that this conjecture holds true for code length $n=p-1$, where $p$ is a prime such that 2 is its primitive root.