The index coding problem provides a simple yet rich model for several important engineering tasks such as satellite communication, content broadcasting, distributed caching, device-to-device relaying, and interference management. This monograph provides a broad overview of this fascinating subject, focusing on the simplest form of multiple-unicast index coding. The main objective in studying the index coding problem are to characterize the capacity region for a general index coding instance in a computable expression and to develop the coding scheme that can achieve it. Despite their simplicity, these two closely related questions are extremely difficult and precise answers to them, after twenty years of vigorous investigation, are still in terra incognita. There are, nonetheless, many elegant results that shed light on the fundamental challenges in multiple-unicast network communication and expose intriguing interplay between coding theory, graph theory, and information theory. This monograph contains a concise survey of these results in a unified framework. It further discusses the relation to Network Coding and Distributed Storage. Fundamentals of Index Coding gives the reader a concise, yet comprehensive, overview of the work undertaken on this important topic; its relationship to adjacent areas and lays the groundwork for future research. It is a valuable starting point for all researchers and students in Information Theory.

/pdf/fundamentals-of-index-coding-2c0j10ry8l.pdf

Fundamentals of Index Coding

We study two basic problems regarding edit errors (insertions and deletions). The first one is document exchange, where two parties Alice and Bob hold two strings x and y with a bounded edit distance k. The goal is to have Alice send a short sketch to Bob, so that Bob can recover x based on y and the sketch. The second one is the fundamental problem of designing error correcting codes for edit errors, where the goal is to construct an explicit code to transmit a message x through a channel that can add at most k worst case insertions and deletions, so that the original message x can be successfully recovered at the other end of the channel. Both problems have been extensively studied for decades, and in this paper we focus on deterministic document exchange protocols and binary codes for insertions and deletions (insdel codes). If the length of x is n, then it is known that for small k (e.g., k ≤ n/4), in both problems the optimal sketch size or the optimal number of redundant bits is Θ(k log n/k). In particular, this implies the existence of binary codes that can correct e fraction of insertions and deletions with rate 1-Θ(e log (1/e). However, known constructions are far from achieving these bounds. In this paper we significantly improve previous results on both problems. For document exchange, we give an efficient deterministic protocol with sketch size O(k log^2 n/k). This significantly improves the previous best known deterministic protocol, which has sketch size O(k^2 + k log^2 n) [2]. For binary insdel codes, we obtain the following results: 1) An explicit binary insdel code which encodes an n-bit message x against k errors with redundancy O(k log^2 n/k). In particular this implies an explicit family of binary insdel codes that can correct e fraction of insertions and deletions with rate 1-O(e log^2 1/(1-e))=1-O(e). This significantly improves the previous best known result which only achieves rate 1-O(√e) [11], [10], and is optimal up to a log (1/e) factor. 1) An explicit binary insdel code which encodes an n-bit message x against k errors with redundancy O(k log n). This significantly improves the previous best known result of [4], which only works for constant k and has redundancy O(k^2 log k log n); and that of [2], which has redundancy O(k^2 + k log^2 n). Our code has optimal redundancy for k ≤ n^1-α, any constant 0 < α < 1. This is the first explicit construction of binary insdel codes that has optimal redundancy for a wide range of error parameters k, and this brings our understanding of binary insdel codes much closer to that of standard binary error correcting codes. In obtaining our results we introduce several new techniques. Most notably, we introduce the notion of e-self matching hash functions and e-synchronization hash functions. We believe our techniques can have further applications in the literature.

Deterministic Document Exchange Protocols, and Almost Optimal Binary Codes for Edit Errors

Thank you very much for downloading algebraic geometric codes. Maybe you have knowledge that, people have search hundreds times for their chosen readings like this algebraic geometric codes, but end up in harmful downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they are facing with some harmful bugs inside their laptop. algebraic geometric codes is available in our digital library an online access to it is set as public so you can get it instantly. Our book servers saves in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the algebraic geometric codes is universally compatible with any devices to read.

Algebraic Geometric Codes

We give the first communication-optimal document exchange protocol. For any n and k < n our randomized scheme takes any n-bit file F and computes a Θ(k log n/k) -bit summary from which one can reconstruct F, with high probability, given a related file F' with edit distance ED(F,F') ≤ k. The size of our summary is information-theoretically order optimal for all values of k, giving a randomized solution to a longstanding open question of [Orlitsky; FOCS'91]. It also is the first non-trivial solution for the interesting setting where a small constant fraction of symbols have been edited, producing an optimal summary of size O(H(δ)n) for k=δ n. This concludes a long series of better-and-better protocols which produce larger summaries for sub-linear values of k and sub-polynomial failure probabilities. In particular, the recent break-through of [Belazzougui, Zhang; FOCS'16] assumes that k < n^e, produces a summary of size O(klog^2 k + k log n), and succeeds with probability 1-(k log n)^ -O(1). We also give an efficient derandomized document exchange protocol with summary size O(k log^2 n/k). This improves, for any k, over a deterministic document exchange protocol by Belazzougui with summary size O(k^2 + k log^2 n). Our deterministic document exchange directly provides new efficient systematic error correcting codes for insertions and deletions. These (binary) codes correct any δ fraction of adversarial insertions/deletions while having a rate of 1 - O(δ log^2 1/δ) and improve over the codes of Guruswami and Li and Haeupler, Shahrasbi and Vitercik which have rate 1 - Θ (√δ log^O(1) 1/e).

Optimal Document Exchange and New Codes for Insertions and Deletions

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 $\epsilon >0$, synchronization strings allow to index a sequence with an $\epsilon^{-O(1)}$ size alphabet such that one can efficiently transform $k$ synchronization errors into $(1+\epsilon)k$ 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: Codes for Insertions and Deletions Approaching the Singleton Bound

We introduce synchronization strings, which provide 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 more commonly considered half-errors, i.e., symbol corruptions and erasures. For every e > 0, synchronization strings allow to index a sequence with an e-O(1) size alphabet such that one can efficiently transform k synchronization errors into (1 + e)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. As Mitzenmacher puts it in his 2009 survey: "Channels with synchronization errors ... are simply not adequately understood by current theory. Given the near-complete knowledge we have for channels with erasures and errors ... our lack of understanding about channels with synchronization errors is truly remarkable." Indeed, it took until 1999 for the first insdel codes with constant rate, constant distance, and constant alphabet size to be constructed and only since 2016 are there constructions of constant rate indel codes for asymptotically large noise rates. Even in the asymptotically large or small noise regime these codes are 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 straight forward 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 only a small increase in the 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, for the complete noise spectrum we obtain efficient "near-MDS" insdel codes which get arbitrarily close to the optimal rate-distance tradeoff given by the Singleton bound. In particular, for any δ ∈ (0,1) and e > 0 we give insdel codes achieving a rate of 1 - ξ - e over a constant size alphabet that efficiently correct a δ fraction of insertions or deletions.

https://dl.acm.org/doi/pdf/10.1145/3055399.3055498

Synchronization strings: codes for insertions and deletions approaching the Singleton bound

We give a complete answer to the following basic question: ”What is the maximal fraction of deletions or insertions tolerable by q-ary list-decodable codes with non-vanishing information rate?” This question has been open even for binary codes, including the restriction to the binary insertion-only setting, where the best-known result was that a γ≤ 0.707 fraction of insertions is tolerable by some binary code family. For any desired є>0, we construct a family of binary codes of positive rate which can be efficiently list-decoded from any combination of γ fraction of insertions and δ fraction of deletions as long as γ+2δ≤ 1−e. On the other hand, for any γ, δ with γ+2δ=1 list-decoding is impossible. Our result thus precisely characterizes the feasibility region of binary list-decodable codes for insertions and deletions. We further generalize our result to codes over any finite alphabet of size q. Surprisingly, our work reveals that the feasibility region for q>2 is not the natural generalization of the binary bound above. We provide tight upper and lower bounds that precisely pin down the feasibility region, which turns out to have a (q−1)-piece-wise linear boundary whose q corner-points lie on a quadratic curve. The main technical work in our results is proving the existence of code families of sufficiently large size with good list-decoding properties for any combination of δ,γ within the claimed feasibility region. We achieve this via an intricate analysis of codes introduced by [Bukh, Ma; SIAM J. Discrete Math; 2014]. Finally, we give a simple yet powerful concatenation scheme for list-decodable insertion-deletion codes which transforms any such (non-efficient) code family (with information rate zero) into an efficiently decodable code family with constant rate.

https://dl.acm.org/doi/pdf/10.1145/3357713.3384262

Optimally resilient codes for list-decoding from insertions and deletions

This paper gives new results for synchronization strings, a powerful combinatorial object introduced by [Haeupler, Shahrasbi; STOC’17] that allows to efficiently deal with insertions and deletions in various communication problems: - We give a deterministic, linear time synchronization string construction, improving over an O(n5) time randomized construction. Independently of this work, a deterministic O(n log2 logn) time construction was proposed by Cheng, Li, and Wu. - We give a deterministic construction of an infinite synchronization string which outputs the first n symbols in O(n) time. Previously it was not known whether such a string was computable. - Both synchronization string constructions are highly explicit, i.e., the ith symbol can be deterministically computed in O(logi) time. - This paper also introduces a generalized notion we call long-distance synchronization strings. Such strings allow for local and very fast decoding. In particular only O(log3 n) time and access to logarithmically many symbols is required to decode any index. The paper also provides several applications for these improved synchronization strings: - For any δ 0 we provide an insdel error correcting block code with rate 1 − δ − є which can correct any δ/3 fraction of insertion and deletion errors in O(n log3 n) 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 local decodability implies that error correcting codes constructed with long-distance synchronization strings can not only efficiently recover from δ fraction of insdel errors but, similar to [Schulman, Zuckerman; TransInf’99], also from any O(δ / logn) fraction of block transpositions and block replications. These block corruptions allow arbitrarily long substrings to be swapped or replicated anywhere. - We show that highly explicitness and local decoding allow for infinite channel simulations with exponentially smaller memory and decoding time requirements. These simulations can then be used to give the first near linear time interactive coding scheme for insdel errors, similar to the result of [Brakerski, Naor; SODA’13] for Hamming errors.

https://dl.acm.org/doi/pdf/10.1145/3188745.3188940

Synchronization strings: explicit constructions, local decoding, and applications

Already in the 1960s, Levenshtein and others studied error-correcting codes that protect against synchronization errors, such as symbol insertions and deletions However, despite significant efforts, progress on designing such codes has been lagging until recently, particularly compared to the detailed understanding of error-correcting codes for symbol substitution or erasure errors This paper surveys the recent progress in designing efficient error-correcting codes over finite alphabets that can correct a constant fraction of worst-case insertions and deletions Most state-of-the-art results for such codes rely on synchronization strings, simple yet powerful pseudo-random objects that have proven to be very effective solutions for coping with synchronization errors in various settings This survey also includes an overview of what is known about synchronization strings and discusses communication settings related to error-correcting codes in which synchronization strings have been applied

Amirbehshad Shahrasbi

Papers

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

Synchronization strings: codes for insertions and deletions approaching the Singleton bound

Optimally resilient codes for list-decoding from insertions and deletions

Synchronization strings: explicit constructions, local decoding, and applications

Synchronization Strings and Codes for Insertions and Deletions—A Survey