On the Unique Decodability of Insertion-Correcting Codes Beyond the Guarantee

Unlike the space of received words generated by substitution errors, the space of received words generated by insertion errors is infinite. Given an arbitrary code, it is possible for there to exist an infinite number of received words that are unique to a particular codeword. This work explores the extent to which an arbitrary insertion-correcting code can take advantage of this fact. Such questions are relevant today, because insertion errors frequently occur in DNA, a medium which is beginning to be used for long term data storage. For a particular codeword c of length n, we are interested in two particular measures. The first is the fraction of received words of length n + t which are unique to c. The second is the fraction of insertion histories of length t which give rise to received words of length n + t that are unique to c. The first measure is the probability of uniquely decoding when t insertions occur, if all possible length n+t received words are equally likely. The second measure is the probability of uniquely decoding when t consecutive insertions occur, and each insertion position and element are selected uniformly at random. This paper attempts to better understand the behavior of these measures for arbitrary insertion correcting codes, placing a particular emphasis on limiting behavior as the number of insertions increases, or the code-length increases. Our most substantial contribution is the derivation of upper bounds on both of these measures, which are mathematically related to Levinshtein’s reconstruction problem. We also establish the positivity of these measures for at least one codeword in every code.