Equivalence of Insertion/Deletion Correcting Codes for d-dimensional Arrays

Abstract

We consider the problem of correcting insertion and deletion errors in the d-dimensional space. This problem is well understood for vectors (one-dimensional space) and was recently studied for arrays (two-dimensional space). For vectors and arrays, the problem is motivated by several practical applications such as DNA-based storage and racetrack memories. From a theoretical perspective, it is interesting to know whether the same properties of insertion/deletion correcting codes generalize to the d-dimensional space. In this work, we show that the equivalence between insertion and deletion correcting codes generalizes to the d-dimensional space. As a particular result, we show the following missing equivalence for arrays: a code that can correct tr and tc row/column deletions can correct any combination of t ins r + t del r = tr and t ins c + t del c = tc row/column insertions and deletions. The fundamental limit on the redundancy and a construction of insertion/deletion correcting codes in the ddimensional space remain open for future work.

Database Matching Under Noisy Synchronization Errors

“…Other natural extensions beyond those studied in this paper include the finite column size regime, where tools from finite-blocklength information theory could be useful, and practical algorithms with theoretical guarantees. An extensive analysis of the parallels between database matching under synchronization errors and two-dimensional synchronization channels [59], [60] and the construction of codes tailored to correct the error patterns investigated in this paper could be an interesting line of future work. Finally, one can extend our adversarial setting into a noisy one where the privacy-preserving mechanism not only deletes columns but also introduces intentional noise on the microdata, and investigate the adversarial matching capacity through a worst-case analysis.…”