Sarit Buzaglo

Error-correcting codes for multipermutations

Multipermutations appear in various applications in information theory. New applications such as rank modulation for flash memories and voting have suggested the need to consider error-correcting codes for multipermutations. The construction of codes is challenging when permutations are considered and it becomes even a harder problem for multipermutations. In this paper we discuss the general problem of error-correcting codes for multipermutations. We present some tight bounds on the size of error-correcting codes for several families of multipermutations. We find the capacity of the channels of multipermutations and characterize families of perfect codes in this metric which we believe are the only such perfect codes.

Bounds on the Size of Permutation Codes With the Kendall -Metric

The rank modulation scheme has been proposed for efficient writing and storing data in nonvolatile memory storage. Error correction in the rank modulation scheme is done by considering permutation codes. In this paper, we consider codes in the set of all permutations on n elements, Sn, using the Kendall τ-metric. The main goal of this paper is to derive new bounds on the size of such codes. For this purpose, we also consider perfect codes, diameter perfect codes, and the size of optimal anticodes in the Kendall τ-metric, structures which have their own considerable interest. We prove that there are no perfect single-error-correcting codes in Sn, where n>4 is a prime or 4≤n≤10 . We present lower bounds on the size of optimal anticodes with odd diameter. As a consequence, we obtain a new upper bound on the size of codes in Sn with even minimum Kendall τ-distance. We present larger single-error-correcting codes than the known ones in S5 and S7.

Tilings With

An n-dimensional chair consists of an n -dimensional box from which a smaller n-dimensional box is removed. A tiling of an n-dimensional chair has two nice applications in some memories using asymmetric codes. The first one is in the design of codes that correct asymmetric errors with limited magnitude. The second one is in the design of n cells q -ary write-once memory codes. We show an equivalence between the design of a tiling with an integer lattice and the design of a tiling from a generalization of splitting (or of Sidon sequences). A tiling of an n -dimensional chair can define a perfect code for correcting asymmetric errors with limited magnitude. We present constructions for such tilings and prove cases where perfect codes for these type of errors do not exist.

n

The rank modulation scheme has been proposed for efficient writing and storing data in non-volatile memory storage. Error-correction in the rank modulation scheme is done by considering permutation codes. In this paper we consider codes in the set of all permutations on n elements, Sn, using the Kendalls τ-metric. We prove that there are no perfect single-error-correcting codes in Sn, where n > 4 is a prime or 4 ≤ n ≤ 10. We also prove that if such a code exists for n which is not a prime then the code should have some uniform structure. We define some variations of the Kendalls τ-metric and consider the related codes and specifically we prove the existence of a perfect single-error-correcting code in S5. Finally, we examine the existence problem of diameter perfect codes in Sn and obtain a new upper bound on the size of a code in Sn with even minimum Kendalls τ-distance.

-Dimensional Chairs and Their Applications to Asymmetric Codes

The goal of this paper is to construct systematic error-correcting codes for permutations and multi-permutations in the Kendalls τ-metric. These codes are important in new applications such as rank modulation for flash memories. The construction is based on error-correcting codes for multi-permutations and a partition of the set of permutations into error-correcting codes. For a given large enough number of information symbols k, and for any integer t, we present a construction for (k + r, k) systematic t-error-correcting codes, for permutations from Sk+r, with less redundancy symbols than the number of redundancy symbols in the codes of the known constructions. In particular, for a given t and for sufficiently large k we can obtain r = t+1. The same construction is also applied to obtain related systematic error-correcting codes for multi-permutations.

Perfect permutation codes with the Kendall's τ-metric

Inter-cell interference (ICI) is a significant cause of errors in flash memories. In single-level cell (SLC) flash memory, ICI arises when 1 0 1 patterns are programmed either in the horizontal or vertical directions. Since data pages are written sequentially in horizontal wordlines, one can mitigate the effects of horizontal ICI by applying conventional constrained codes that forbid the 1 0 1 pattern. This approach does not address the problem of vertical ICI, however. In this paper, a row-by-row coding technique that eliminates vertical 1 0 1 patterns while preserving the sequential wordline programming order is presented. This scheme, though efficient, necessarily suffers a rate loss of almost 20%. We therefore propose another coding scheme, combining a weak constraint on vertical 1 0 1 patterns with a systematic error-correcting code, that can mitigate vertical ICI errors while achieving a higher overall coding rate, provided that the vertical ICI error probability is sufficiently small. Some extensions for multi-level cell (MLC) flash memory are discussed as well.

Systematic codes for rank modulation

The existence question for tiling of the

Coding schemes for inter-cell interference in flash memory

n

Tilings by

-dimensional Euclidian space by crosses is well known. A few existence and nonexistence results are known in the literature. Of special interest are tilings of the Euclidian space by crosses with arms of length one, also known as Lee spheres with radius one. Such a tiling forms a perfect code. In this paper crosses with arms of length half are considered. These crosses are scaled by two to form a discrete shape. A tiling with this shape is also known as a perfect dominating set. We prove that an integer tiling for such a shape exists if and only if

(0.5,n)

n=2^t-1