Theoretical Bounds and Constructions of Codes in the Generalized Cayley Metric

TL;DR

This paper investigates the theoretical limits and provides new explicit constructions for permutation codes in the generalized Cayley metric, which models complex error types in applications like storage and genomics.

Contribution

It introduces systematic, explicit code constructions in the generalized Cayley metric and proves their order-optimality and reduced redundancy compared to existing methods.

Findings

Proposed an order-optimal, non-constructive coding scheme.

Developed explicit, systematic code constructions.

Codes have less redundancy for large codelengths and small errors.

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. Relying on the breakpoint analysis proposed by Chee and Vu, we first propose a coding scheme that is order-optimal albeit not constructive based on this method. We then develop another construction of permutation codes in the generalized Cayley distance. This scheme is both explicit and systematic. We also prove the existence of order-optimal systematic codes and offer a concrete construction based on this method. For the generalized Cayley metric, we prove that our coding schemes have less redundancy than the existing codes based on interleaving when the codelength is sufficiently large and the number of errors is relatively small.