Breakpoint analysis and permutation codes in generalized Kendall tau and Cayley metrics

Theoretical Bounds and Constructions of Codes in the Generalized Cayley Metric

“…A generalized transposition φ (i 1 , j 1 , i 2 , j 2 ) ∈ S N , where i 1 ≤ j 1 < i 2 ≤ j 2 ∈ [N ], refers to a permutation that is obtained from swapping two segments, e [i 1 , j 1 ] and e [i 2 , j 2 ], of the identity permutation [5], φ (i 1 , j 1 , i 2 , j 2 ) (1, · · · , i 1 − 1, i 2 , · · · , j 2 , j 1 + 1, · · · , i 2 − 1, i 1 , · · · , j 1 , j 2 + 1, · · · , N ) .…”

“…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).…”

“…Definition 4. The block permutation weight w B (π) is defined as the number of consecutive pairs in π that do not belong to A(e) (w B is exactly the number of so-called breakpoints in [5]), i.e., w B (π) |A(π) \ A(e)|.…”

“…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. Codes in the generalized Cayley metric were first studied in [5] using the breakpoint analysis, wherein a coding scheme is constructed based on permutation codes, previously introduced in [10], in the Ulam metric. Let N be the length of the codewords, and t be the maximum number of errors in the generalized Cayley metric.…”

“…Let N be the length of the codewords, and t be the maximum number of errors in the generalized Cayley metric. While the coding scheme proposed in [5] is explicitly constructive and implementable, the interleaving technique used inevitably incurs a noticeable redundancy of Θ (N ), without even considering the number of errors that the code is able to correct. As we show later, the best possible redundancy of a length-N code that corrects t generalized transposition errors is Θ (t log N ).…”