Optimal Codes Detecting Deletions in Concatenated Binary Strings Applied to Trace Reconstruction

Abstract

Consider two or more strings $\mathbf{x}^1,\mathbf{x}^2,\ldots,$ that are concatenated to form $\mathbf{x}=\langle \mathbf{x}^1,\mathbf{x}^2,\ldots \rangle$. Suppose that up to $\delta$ deletions occur in each of the concatenated strings. Since deletions alter the lengths of the strings, a fundamental question to ask is: how much redundancy do we need to introduce in $\mathbf{x}$ in order to recover the boundaries of $\mathbf{x}^1,\mathbf{x}^2,\ldots$? This boundary problem is equivalent to the problem of designing codes that can detect the exact number of deletions in each concatenated string. In this work, we answer the question above by first deriving converse results that give lower bounds on the redundancy of deletion-detecting codes. Then, we present a marker-based code construction whose redundancy is asymptotically optimal in $\delta$ among all families of deletion-detecting codes, and exactly optimal among all block-by-block decodable codes. To exemplify the usefulness of such deletion-detecting codes, we apply our code to trace reconstruction and design an efficient coded reconstruction scheme that requires a constant number of traces.