8,300 research outputs found

    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

    Importance of Symbol Equity in Coded Modulation for Power Line Communications

    Full text link
    The use of multiple frequency shift keying modulation with permutation codes addresses the problem of permanent narrowband noise disturbance in a power line communications system. In this paper, we extend this coded modulation scheme based on permutation codes to general codes and introduce an additional new parameter that more precisely captures a code's performance against permanent narrowband noise. As a result, we define a new class of codes, namely, equitable symbol weight codes, which are optimal with respect to this measure

    Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes

    Get PDF
    We present two methods for the construction of quantum circuits for quantum error-correcting codes (QECC). The underlying quantum systems are tensor products of subsystems (qudits) of equal dimension which is a prime power. For a QECC encoding k qudits into n qudits, the resulting quantum circuit has O(n(n-k)) gates. The running time of the classical algorithm to compute the quantum circuit is O(n(n-k)^2).Comment: 18 pages, submitted to special issue of IJFC
    corecore