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We investigate the list decodability of symbol-pair codes in this paper. First, we show that the list decodability of every symbol-pair code does not exceed the Gilbert–Varshamov bound. On the other hand, we are able to prove that with high probability, a random symbol-pair code can be list decoded up to the Gilbert–Varshamov bound. Our second result of this paper is to derive the Johnson-type bound, i.e., a lower bound on list decoding radius in terms of minimum distance. Finally, we present a list decoding algorithm of Reed–Solomon codes beyond the Johnson-type bound in the pair metric. 1 A symbol-pair code is referred to a code in the pair metric. [ABSTRACT FROM AUTHOR]

Symbol-pair read channels, in which the outputs of the read process are pairs of consecutive symbols, were recently studied by Cassuto and Blaum. This new paradigm is motivated by the limitations of the reading process in high density data storage systems. They studied error correction in this new paradigm, specifically, the relationship between the minimum Hamming distance of an error correcting code and the minimum pair distance, which is the minimum Hamming distance between symbol-pair vectors derived from codewords of the code. It was proved that for a linear cyclic code with minimum Hamming distance dH , the corresponding minimum pair distance is at least dH+3 . In this paper, we show that, for a given linear cyclic code with a minimum Hamming distance dH , the minimum pair distance is at least dH +\left \lceil{ {dH/{2}}}\right \rceil . We then describe a decoding algorithm, based upon a bounded distance decoder for the cyclic code, whose symbol-pair error correcting capabilities reflect the larger minimum pair distance. Finally, we consider the case where the read channel output is a larger number, $b \geqslant 3$ , of consecutive symbols, and we provide extensions of several concepts, results, and code constructions to this setting. [ABSTRACT FROM AUTHOR]

In this paper, we propose two new generic attacks on the rank syndrome decoding (RSD) problem. Let C be a random [n,k] rank code over GF(q^m) and let y=x+e be a received word, such that x \in C and rank (e)=r . The first attack, the support attack, is combinatorial and permits to recover an error e of rank weight r operations on GF(q) . This new attack improves the exponent for the best generic attack for the RSD problem in the case n > m , by introducing the ratio m/n in the exponential coefficient of the previously best known attacks. The second attack, the annulator polynomial attack, is an algebraic attack based on the theory of q rather than in GF(q) as it is usually the case. We consider two approaches to solve the problem in this new setting. The linearization technique shows that if n \ge (k+1) (r+1)-1 the RSD problem can be solved in polynomial time. More generally, we prove that if , the RSD problem can be solved with an average complexity of O(r^3k^3q^r\lceil (((r+1)(k+1)-(n+1))/r) \rceil ) operations in the base field GF(q) . We also consider solving with Gröbner bases for which we discuss theoretical complexity, we also consider hybrid solving with Gröbner bases on practical parameters. As an example of application, we use our new attacks on all recent cryptosystems parameters, which repair the GPT cryptosystem, we break all examples of published proposed parameters, and some parameters are broken in less than 1 s in certain cases. [ABSTRACT FROM AUTHOR]

Error-correcting codes for permutations have received considerable attention in the past few years, especially in applications of the rank modulation scheme for flash memories. While codes over several metrics have been studied, such as the Kendall $\tau $ , Ulam, and Hamming distances, no recent research has been carried out for erasures and deletions over permutations. In rank modulation, flash memory cells represent a permutation, which is induced by their relative charge levels. We explore problems that arise when some of the cells are either erased or deleted. In each case, we study how these erasures and deletions affect the information carried by the remaining cells. In particular, we study models that are symbol-invariant, where unaffected elements do not change their corresponding values from those in the original permutation, or permutation-invariant, where the remaining symbols are modified to form a new permutation with fewer elements. Our main approach in tackling these problems is to build upon the existing works of error-correcting codes and leverage them in order to construct codes in each model of deletions and erasures. The codes we develop are in certain cases asymptotically optimal, while in other cases, such as for codes in the Ulam distance, improve upon the state of the art results. [ABSTRACT FROM PUBLISHER]

We consider rank modulation codes for flash memories that allow for handling arbitrary charge-drop errors. Unlike classical rank modulation codes used for correcting errors that manifest themselves as swaps of two adjacently ranked elements, the proposed translocation rank codes account for more general forms of errors that arise in storage systems. Translocations represent a natural extension of the notion of adjacent transpositions and as such may be analyzed using related concepts in combinatorics and rank modulation coding. Our results include derivation of the asymptotic capacity of translocation rank codes, construction techniques for asymptotically good codes, as well as simple decoding methods for one class of constructed codes. As part of our exposition, we also highlight the close connections between the new code family and permutations with short common subsequences, deletion and insertion error-correcting codes for permutations, and permutation codes in the Hamming distance. [ABSTRACT FROM PUBLISHER]

A new coding framework is established for channels whose outputs are overlapping pairs of symbols. Such channels are motivated by storage applications in which the spatial resolution of the reader may be insufficient to isolate adjacent symbols. Reading symbols as pairs changes the coding-theoretic error model from the standard bounded number of symbol errors to a bounded number of pair errors. Starting from the most basic coding-theoretic questions, the paper studies codes that protect against pair-errors. It provides answers on pair-error correctability conditions, code construction and decoding, and lower and upper bounds on code sizes. Asymptotic analysis of pair-error correction shows that there exist pair-error codes with rates that are strictly higher than the best known codes in the Hamming metric. [ABSTRACT FROM AUTHOR]

A well-known conjecture of Golomb and Welch is that the only nontrivial perfect codes in the Lee and Manhattan metrics have length two or minimum distance three. This problem and related topics were subject for extensive research in the last 40 years. In this paper, two product constructions for perfect Lee codes and diameter perfect Lee codes are presented. These constructions yield a large number of nonlinear perfect codes and nonlinear diameter perfect codes in the Lee and Manhattan metrics. A short survey and other related problems on perfect codes in the Lee and Manhattan metrics are also discussed. [ABSTRACT FROM AUTHOR]