Systematic Error-Correcting Codes for Permutations and Multi-Permutations

Theoretical Bounds and Constructions of Codes in the Generalized Cayley Metric

“…Example 2. Let π 1 = (3,5,6,7,9,8,1,2,10,4), π 2 = (3, 1, 2, 8,5,6,7,9,10,4). Define ψ i , 1 ≤ i ≤ 4, and σ as follows, 5,6,7,9), ψ 3 = (8), ψ 4 = (1, 2), ψ 5 = (10, 4), σ = (1, 4, 3, 2, 5).…”

“…Example 6. Suppose π = (1, 4, 5, 7, 6, 2, 3), I = (4, 1, 2, 2), then E(π, I) = (1,9,4,8,5,7,6,2,11,10,3) .…”

“…Permutation codes in the Kendall-τ metric and the Ulam metric, along with codes in the Levenshtein metric have been recently actively studied, in [6]- [8], [9]- [11], and [12], [13], respectively. Generalized transposition errors subsume transpositions and translocations that the Kendall-τ metric and Ulam metric capture, and in particular no restrictions are imposed on the positions and lengths of the translocated segments as in these two metrics.…”

New theoretical bounds and constructions of permutation codes under block permutation metric

“…Example 2.6. Let π 1 = (4, 8, 3, 2, 6, 7, 5, 1, 9), π 2 = (6, 7, 8, 3, 2, 5, 1,9,4). Their characteristic sets are A(π 1 ) = {(4, 8), (8,3), (3,2), (2,6), (6, 7), (7,5), (5,1), (1,9)}, A(π 2 ) = {(6, 7), (7,8), (8,3), (3,2), (2,5), (5,1), (1,9), (9, 4)}, and thus we have d B (π 1 , π 2 ) = |A (π 1 ) \ A (π 2 )| = |{(4, 8), (2, 6), (7,5)…”

“…Permutation codes under several different metrics are widely used due to their various applications. Especially in recent years, permutation codes under Kendall's τ -metric, Ulam metric and Cayley metric have been extensively studied in clouding storage systems, genome resequencing and the rank modulation scheme of flash memories [3], [4], [6], [8], [9], [11], [15], [17]. Under these metrics, codes are designed to correct transposition errors or translocation errors.…”