Decoding Frequency Permutation Arrays Under Chebyshev Distance

Abstract

A frequency permutation array (FPA) of length = and distance is a set of permutations on a multiset over symbols, where each symbol appears exactly times and the distance between any two elements in the array is at least . FPA generalizes the notion of permutation array. In this paper, under the Chebyshev distance, we first prove lower and upper bounds on the size of FPA. Then we give several constructions of FPAs, and some of them come with efficient encoding and decoding capabilities. Moreover, we show one of our designs is locally decodable, i.e., we can decode a message bit by reading at most + 1 symbols, which has an interesting application to private information retrieval.

Multipermutation Codes in the Ulam Metric for Nonvolatile Memories

“…These works were motivated by different considerations -the former aiming to increase the number of possible re-writes between block erasures, and the latter focusing on the advantages of multipermutation coding with respect to cell leakage, over-injection issues, and charge fluctuations. In addition, multipermutation codes were also recently reported for the Chebyshev distance in [18], [19] and for the Kendall τ distance in [20], [21].…”

“…One of the key ingredients of our constructions is permutation interleaving, which we proposed for Ulam metric code design in [7]. The related idea of restricting certain positions in the codewords to certain values was first described in [11], [12], while interleaving in the Chebyshev metric was discussed in [18].…”

“…The code C, constructed according to (18) contains ) and consider α = (4, 1, 5, 2, 3, 6) ∈ R 2 (π) and β = (4, 3, 5, 2, 6, 1) ∈ R 2 (σ). It can be observed that α P1 = π P1 = (1, 2, 3) and α P2 = π P2 = (4, 5, 6).…”

“…Multipermutation codes in the Hamming metric, known as constant composition codes or frequency permutation arrays (FPAs), were studied in several papers including [13]- [16]. For nonvolatile memories, multipermutation coding was proposed by En Gad et al [17], as well as by Shieh and Tsai [18]. These works were motivated by different considerations -the former aiming to increase the number of possible re-writes between block erasures, and the latter focusing on the advantages of multipermutation coding with respect to cell leakage, over-injection issues, and charge fluctuations.…”

Limited-magnitude error-correcting Gray codes for rank modulation

“…Several error models have been studied for rank modulation, including the Kendall τ-metric [2], [22], [25], [39], the ℓ ∞ -metric [24], [30], [32], [33] and other examples [11], [16]. In this paper we focus on the ℓ ∞ -metric, which models limitedmagnitude or spike noise, i.e., we assume that the rank of any given cell-its position when sorting the group of cells-could not have changed by more than a given amount.…”