Abstract:Abstract-We address the problem of multipermutation code design in the Ulam metric for novel storage applications. Multipermutation codes are suitable for flash memory where cell charges may share the same rank. Changes in the charges of cells manifest themselves as errors whose effects on the retrieved signal may be measured via the Ulam distance. As part of our analysis, we study multipermutation codes in the Hamming metric, known as constant composition codes. We then present bounds on the size of multiperm… Show more
Permutation codes under different metrics have been extensively studied due to their potentials in various applications. Generalized Cayley metric is introduced to correct generalized transposition errors, including previously studied metrics such as Kendall's τ -metric, Ulam metric and Cayley metric as special cases. Since the generalized Cayley distance between two permutations is not easily computable, Yang et al. introduced a related metric of the same order, named the block permutation metric. Given positive integers n and d, let C B (n, d) denote the maximum size of a permutation code in S n with minimum block permutation distance d. In this paper, we focus on the theoretical bounds of C B (n, d) and the constructions of permutation codes under block permutation metric. Using a graph theoretic approach, we improve the Gilbert-Varshamov type bound by a factor of Ω(log n), when d is fixed and n goes into infinity. We also propose a new encoding scheme based on binary constant weight codes. Moreover, an upper bound beating the sphere-packing type bound is given when d is relatively close to n.
Permutation codes under different metrics have been extensively studied due to their potentials in various applications. Generalized Cayley metric is introduced to correct generalized transposition errors, including previously studied metrics such as Kendall's τ -metric, Ulam metric and Cayley metric as special cases. Since the generalized Cayley distance between two permutations is not easily computable, Yang et al. introduced a related metric of the same order, named the block permutation metric. Given positive integers n and d, let C B (n, d) denote the maximum size of a permutation code in S n with minimum block permutation distance d. In this paper, we focus on the theoretical bounds of C B (n, d) and the constructions of permutation codes under block permutation metric. Using a graph theoretic approach, we improve the Gilbert-Varshamov type bound by a factor of Ω(log n), when d is fixed and n goes into infinity. We also propose a new encoding scheme based on binary constant weight codes. Moreover, an upper bound beating the sphere-packing type bound is given when d is relatively close to n.
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