Bounds on the Size of Permutation Codes with the Kendall $\tau$-Metric

TL;DR

This paper establishes new bounds on permutation codes under the Kendall tau-metric, proving the non-existence of perfect single-error-correcting codes for certain sizes, and constructing larger codes in specific cases.

Contribution

It introduces new bounds on code sizes, proves the non-existence of perfect codes for certain parameters, and constructs larger single-error-correcting codes in specific permutation groups.

Findings

No perfect single-error-correcting codes in S_n for n>4 prime or 4≤n≤10.

New upper bounds on code sizes with even minimum Kendall tau-distance.

Larger single-error-correcting codes constructed in S_5 and S_7.

Abstract

The rank modulation scheme has been proposed for efficient writing and storing data in non-volatile memory storage. Error-correction in the rank modulation scheme is done by considering permutation codes. In this paper we consider codes in the set of all permutations on $n$ elements, $S_n$, using the Kendall $\tau$-metric. The main goal of this paper is to derive new bounds on the size of such codes. For this purpose we also consider perfect codes, diameter perfect codes, and the size of optimal anticodes in the Kendall $\tau$-metric, structures which have their own considerable interest. We prove that there are no perfect single-error-correcting codes in $S_n$, where $n>4$ is a prime or $4\leq n\leq 10$. We present lower bounds on the size of optimal anticodes with odd diameter. As a consequence we obtain a new upper bound on the size of codes in $S_n$ with even minimum Kendall $\tau$-distance. We present larger single-error-correcting codes than the known ones in $S_5$ and $S_7$.