Robust Indexing - Optimal Codes for DNA Storage

Abstract

Encoding data as a set of unordered strings is receiving great attention as it captures one of the basic features of DNA storage systems. However, the challenge of constructing optimal redundancy codes for this channel remained elusive. In this paper, we address this problem and present an order-wise optimal construction of codes that are capable of correcting multiple substitution, deletion, and insertion errors for this channel model. The key ingredient in the code construction is a technique we call robust indexing: simultaneously assigning indices to unordered strings (hence, creating order) and also embedding information in these indices. The encoded indices are resilient to substitution, deletion, and insertion errors, and therefore, so is the entire code.

A Practical Indexing Scheme for Noisy Shuffling Channels Using Cosets of Polar Codes

“…The receiver may take advantage of these properties to first decode the noisy anchors, and then employ the retrieved anchors to restore the transmission order [14]. In [15], indexes are encoded by an error correcting code with a minimum distance of 2K + 1, where K is the maximum number of substitution errors imposed by the channel. This approach enables the receiver to recover the indexes using a minimum distance decoder.…”

On Codes for the Noisy Substring Channel

“…To overcome these effects, one approach in existing literature is to add redundancy in the form of indexing, in order to recover the order of substrings (see, e.g., [11]- [13]). A different approach, potentially more applicable to in-vivo DNA storage, is to add redundancy in the form of constraints on the long information string, such that it can be uniquely reconstructed by knowledge of its substrings of a given length (or range of lengths).…”