Optimality conjecture for the binary Varshamov–Tenengolts codes VT_0(n)

Let $A(n,1)$ denote the maximum size of a binary single-deletion-correcting code of length $n$, and let $VT_0(n)$ be the binary Varshamov–Tenengolts code with syndrome $0$. A code is optimal when its size is $A(n,1)$. Optimality conjecture. The codes $VT_0(n)$ are optimal for every $n$, that is, $|VT_0(n)|=A(n,1)$. The conjecture proposes that the standard zero-syndrome Varshamov–Tenengolts code attains the largest possible size among all binary single-deletion-correcting codes of each length. The preceding discussion establishes asymptotic near-optimality and notes that all $VT_i(n)$ are perfect, but it does not establish optimality for every $n$.