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Register a new userMultiple Criss-Cross Insertion and Deletion Correcting Codes
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Lorenz Welter
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper investigates the problem of correcting multiple criss-cross insertions and deletions in arrays. More precisely, we study the unique recovery of $n times n$ arrays affected by $t$-criss-cross deletions defined as any combination of $t_r$ row and $t_c$ column deletions such that $t_r + t_c = t$ for a given $t$. We show an equivalence between correcting $t$-criss-cross deletions and $t$-criss-cross insertions and show that a code correcting $t$-criss-cross insertions/deletions has redundancy at least $tn + t log n - log(t!)$. Then, we present an existential construction of $t$-criss-cross insertion/deletion correcting code with redundancy bounded from above by $tn + mathcal{O}(t^2 log^2 n)$. The main ingredients of the presented code construction are systematic binary $t$-deletion correcting codes and Gabidulin codes. The first ingredient helps locating the indices of the inserted/deleted rows and columns, thus transforming the insertion/deletion-correction problem into a row/column erasure-correction problem which is then solved using the second ingredient.
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In this paper motivated from subspace coding we introduce subspace-metric and subset-metric codes. These are coordinate-position independent pseudometrics and suitable for the folded codes introduced by Guruswami and Rudra. The half-Singleton upper bounds for linear subspace-metric and subset-metric codes are proved. Subspace distances and subset distances of codes are natural lower bounds for insdel distances of codes, and then can be used to lower bound the insertion-deletion error-correcting capabilities of codes. The problem to construct efficient insertion-deletion error-correcting codes is notorious difficult and has attracted a long-time continuous efforts. The recent breakthrough is the algorithmic construction of near-Singleton optimal rate-distance tradeoff insertion-deletion code families by B. Haeupler and A. Shahrasbi in 2017 from their synchronization string technique. However most nice codes in these recent results are not explicit though many of them can be constructed by highly efficient algorithms. Our subspace-metric and subset-metric codes can be used to construct systemic explicit well-structured insertion-deletion codes. We present some near-optimal subspace-metric and subset-metric codes from known constant dimension subspace codes. By analysing the subset distances of folded codes from evaluation codes of linear mappings, we prove that they have high subset distances and then are explicit good insertion-deletion codes
Motivated by applications in machine learning and archival data storage, we introduce function-correcting codes, a new class of codes designed to protect a function evaluation of the data against errors. We show that function-correcting codes are equivalent to irregular-distance codes, i.e., codes that obey some given distance requirement between each pair of codewords. Using these connections, we study irregular-distance codes and derive general upper and lower bounds on their optimal redundancy. Since these bounds heavily depend on the specific function, we provide simplified, suboptimal bounds that are easier to evaluate. We further employ our general results to specific functions of interest and compare our results to standard error-correcting codes which protect the whole data.
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We consider network coding for networks experiencing worst-case bit-flip errors, and argue that this is a reasonable model for highly dynamic wireless network transmissions. We demonstrate that in this setup prior network error-correcting schemes can be arbitrarily far from achieving the optimal network throughput. We propose a new metric for errors under this model. Using this metric, we prove a new Hamming-type upper bound on the network capacity. We also show a commensurate lower bound based on GV-type codes that can be used for error-correction. The codes used to attain the lower bound are non-coherent (do not require prior knowledge of network topology). The end-to-end nature of our design enables our codes to be overlaid on classical distributed random linear network codes. Further, we free internal nodes from having to implement potentially computationally intensive link-by-link error-correction.
Recent advances in DNA sequencing technology and DNA storage systems have rekindled the interest in deletion channels. Multiple recent works have looked at variants of sequence reconstruction over a single and over multiple deletion channels, a notoriously difficult problem due to its highly combinatorial nature. Although works in theoretical computer science have provided algorithms which guarantee perfect reconstruction with multiple independent observations from the deletion channel, they are only applicable in the large blocklength regime and more restrictively, when the number of observations is also large. Indeed, with only a few observations, perfect reconstruction of the input sequence may not even be possible in most cases. In such situations, maximum likelihood (ML) and maximum aposteriori (MAP) estimates for the deletion channels are natural questions that arise and these have remained open to the best of our knowledge. In this work, we take steps to answer the two aforementioned questions. Specifically: 1. We show that solving for the ML estimate over the single deletion channel (which can be cast as a discrete optimization problem) is equivalent to solving its relaxation, a continuous optimization problem; 2. We exactly compute the symbolwise posterior distributions (under some assumptions on the priors) for both the single as well as multiple deletion channels. As part of our contributions, we also introduce tools to visualize and analyze error events, which we believe could be useful in other related problems concerning deletion channels.
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