A comparison between single asymmetric and single insertion/deletion correcting codes using group-theory

The Varshamov-Tenengol´ts codes are binary number-theoretic codes that can correct a single asymmetric error. Levenshtein realised that this same construction can correct a single insertion/deletion error. And yet asymmetric and insertion/deletion errors appear to be fundamentally different. This paper explores this scenario. Constantin and Rao showed that any Abelian group of order n + 1 can correct a single asymmetric error. In this paper, we show that only a subset of Abelian groups can correct a single insertion/deletion error.