Theoretical Bounds and Constructions of Codes in the Generalized Cayley Metric

Abstract

Permutation codes have recently garnered substantial research interest due to their potential in various applications, including cloud storage systems, genome resequencing, and flash memories. In this paper, we study the theoretical bounds and constructions of permutation codes in the generalized Cayley metric. The generalized Cayley metric captures the number of generalized transposition errors in a permutation, and subsumes previously studied error types, including transpositions and translocations, without imposing restrictions on the lengths and positions of the translocated segments. Based on the socalled breakpoint analysis method proposed by Chee and Vu, we first present a coding framework that leads to order-optimal constructions, thus improving upon the existing constructions that are not order-optimal. We then use this framework to also develop an order-optimal coding scheme that is additionally explicit and systematic.

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

“…The constraint of σ being minimal indicates that d B (π 1 , π 2 ) = d if and only if d + 1 is the minimum number of segments that π 1 needs to be divided into for such an operation. This definition is somehow not intuitive enough and thus Yang et al [16] found another way to characterize the block permutation distance explicitly by the characteristic set of a permutation. Definition 2.3.…”

“…The block permutation metric can be characterized by the characteristic set and then some basic properties of the metric can be derived. These are summarized in the following two lemmas proposed in [16]. Lemma 2.4.…”

“…[16] For given integers n and t, t n − √ n − 1, denote the size of a t-block permutation ball as |b B (n, t)|, then we have Lemma 2.10. [16] For given integers n and t, let d = 2t + 1, then we can bound C B (n, d) as…”

“…However, the generalized Cayley distance between two permutations is in general not easily computable and thus the construction of codes is difficult. In [16], Yang et al introduced the block permutation metric which could be simply computed and is of the same order as the generalized Cayley metric. By the metric embedding method, the problem of constructing codes in the generalized Cayley metric is transformed into constructing codes in the block permutation metric.…”

“…By the metric embedding method, the problem of constructing codes in the generalized Cayley metric is transformed into constructing codes in the block permutation metric. Several theoretical bounds (Gilbert-Varshamov type and sphere-packing type) and constructions of codes under block permutation metric are shown in [16].…”