2,945 research outputs found

    Synchronization Strings: Codes for Insertions and Deletions Approaching the Singleton Bound

    Full text link
    We introduce synchronization strings as a novel way of efficiently dealing with synchronization errors, i.e., insertions and deletions. Synchronization errors are strictly more general and much harder to deal with than commonly considered half-errors, i.e., symbol corruptions and erasures. For every ϵ>0\epsilon >0, synchronization strings allow to index a sequence with an ϵO(1)\epsilon^{-O(1)} size alphabet such that one can efficiently transform kk synchronization errors into (1+ϵ)k(1+\epsilon)k half-errors. This powerful new technique has many applications. In this paper, we focus on designing insdel codes, i.e., error correcting block codes (ECCs) for insertion deletion channels. While ECCs for both half-errors and synchronization errors have been intensely studied, the later has largely resisted progress. Indeed, it took until 1999 for the first insdel codes with constant rate, constant distance, and constant alphabet size to be constructed by Schulman and Zuckerman. Insdel codes for asymptotically large or small noise rates were given in 2016 by Guruswami et al. but these codes are still polynomially far from the optimal rate-distance tradeoff. This makes the understanding of insdel codes up to this work equivalent to what was known for regular ECCs after Forney introduced concatenated codes in his doctoral thesis 50 years ago. A direct application of our synchronization strings based indexing method gives a simple black-box construction which transforms any ECC into an equally efficient insdel code with a slightly larger alphabet size. This instantly transfers much of the highly developed understanding for regular ECCs over large constant alphabets into the realm of insdel codes. Most notably, we obtain efficient insdel codes which get arbitrarily close to the optimal rate-distance tradeoff given by the Singleton bound for the complete noise spectrum

    Synchronization Strings: Explicit Constructions, Local Decoding, and Applications

    Full text link
    This paper gives new results for synchronization strings, a powerful combinatorial object that allows to efficiently deal with insertions and deletions in various communication settings: \bullet We give a deterministic, linear time synchronization string construction, improving over an O(n5)O(n^5) time randomized construction. Independently of this work, a deterministic O(nlog2logn)O(n\log^2\log n) time construction was just put on arXiv by Cheng, Li, and Wu. We also give a deterministic linear time construction of an infinite synchronization string, which was not known to be computable before. Both constructions are highly explicit, i.e., the ithi^{th} symbol can be computed in O(logi)O(\log i) time. \bullet This paper also introduces a generalized notion we call long-distance synchronization strings that allow for local and very fast decoding. In particular, only O(log3n)O(\log^3 n) time and access to logarithmically many symbols is required to decode any index. We give several applications for these results: \bullet For any δ0\delta0 we provide an insdel correcting code with rate 1δϵ1-\delta-\epsilon which can correct any O(δ)O(\delta) fraction of insdel errors in O(nlog3n)O(n\log^3n) time. This near linear computational efficiency is surprising given that we do not even know how to compute the (edit) distance between the decoding input and output in sub-quadratic time. We show that such codes can not only efficiently recover from δ\delta fraction of insdel errors but, similar to [Schulman, Zuckerman; TransInf'99], also from any O(δ/logn)O(\delta/\log n) fraction of block transpositions and replications. \bullet We show that highly explicitness and local decoding allow for infinite channel simulations with exponentially smaller memory and decoding time requirements. These simulations can be used to give the first near linear time interactive coding scheme for insdel errors

    Deletion codes in the high-noise and high-rate regimes

    Get PDF
    The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst-case deletions, with a focus on constructing efficiently decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following trade-offs (for any eps > 0): (1) Codes that can correct a fraction 1-eps of deletions with rate poly(eps) over an alphabet of size poly(1/eps); (2) Binary codes of rate 1-O~(sqrt(eps)) that can correct a fraction eps of deletions; and (3) Binary codes that can be list decoded from a fraction (1/2-eps) of deletions with rate poly(eps) Our work is the first to achieve the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets, and correcting a constant fraction of bit deletions with rate aproaching 1. The above results bring our understanding of deletion code constructions in these regimes to a similar level as worst-case errors

    Parsing a sequence of qubits

    Full text link
    We develop a theoretical framework for frame synchronization, also known as block synchronization, in the quantum domain which makes it possible to attach classical and quantum metadata to quantum information over a noisy channel even when the information source and sink are frame-wise asynchronous. This eliminates the need of frame synchronization at the hardware level and allows for parsing qubit sequences during quantum information processing. Our framework exploits binary constant-weight codes that are self-synchronizing. Possible applications may include asynchronous quantum communication such as a self-synchronizing quantum network where one can hop into the channel at any time, catch the next coming quantum information with a label indicating the sender, and reply by routing her quantum information with control qubits for quantum switches all without assuming prior frame synchronization between users.Comment: 11 pages, 2 figures, 1 table. Final accepted version for publication in the IEEE Transactions on Information Theor

    Guess & Check Codes for Deletions and Synchronization

    Full text link
    We consider the problem of constructing codes that can correct δ\delta deletions occurring in an arbitrary binary string of length nn bits. Varshamov-Tenengolts (VT) codes can correct all possible single deletions (δ=1)(\delta=1) with an asymptotically optimal redundancy. Finding similar codes for δ2\delta \geq 2 deletions is an open problem. We propose a new family of codes, that we call Guess & Check (GC) codes, that can correct, with high probability, a constant number of deletions δ\delta occurring at uniformly random positions within an arbitrary string. The GC codes are based on MDS codes and have an asymptotically optimal redundancy that is Θ(δlogn)\Theta(\delta \log n). We provide deterministic polynomial time encoding and decoding schemes for these codes. We also describe the applications of GC codes to file synchronization.Comment: Accepted in ISIT 201

    Non-asymptotic Upper Bounds for Deletion Correcting Codes

    Full text link
    Explicit non-asymptotic upper bounds on the sizes of multiple-deletion correcting codes are presented. In particular, the largest single-deletion correcting code for qq-ary alphabet and string length nn is shown to be of size at most qnq(q1)(n1)\frac{q^n-q}{(q-1)(n-1)}. An improved bound on the asymptotic rate function is obtained as a corollary. Upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length, and this idea is demonstrated by application to sets of run-length limited strings. The problem of finding the largest deletion correcting code is modeled as a matching problem on a hypergraph. This problem is formulated as an integer linear program. The upper bound is obtained by the construction of a feasible point for the dual of the linear programming relaxation of this integer linear program. The non-asymptotic bounds derived imply the known asymptotic bounds of Levenshtein and Tenengolts and improve on known non-asymptotic bounds. Numerical results support the conjecture that in the binary case, the Varshamov-Tenengolts codes are the largest single-deletion correcting codes.Comment: 18 pages, 4 figure
    corecore