Nonexistence of perfect permutation codes under the Kendall \tau-metric

TL;DR

This paper proves the nonexistence of perfect permutation codes under the Kendall tau-metric for various parameters, advancing understanding in coding theory for flash memory applications.

Contribution

It establishes the nonexistence of perfect permutation codes under the Kendall tau-metric for multiple error-correcting scenarios, answering an open problem.

Findings

No perfect codes for t=2,3,4,5 in S_n under Kendall tau-metric.

Recursive formulas for ball sizes in permutation space.

Extended nonexistence results for larger n and t values.

Abstract

In the rank modulation scheme for flash memories, permutation codes have been studied. In this paper, we study perfect permutation codes in $S_n$, the set of all permutations on $n$ elements, under the Kendall \tau-Metric. We answer one open problem proposed by Buzaglo and Etzion. That is, proving the nonexistence of perfect codes in $S_n$, under the Kendall \tau-metric, for more values of $n$. Specifically, we present the recursive formulas for the size of a ball with radius $r$ in $S_n$ under the Kendall \tau-metric. Further, We prove that there are no perfect $t$-error-correcting codes in $S_n$ under the Kendall $\tau$-metric for some $n$ and $t$=2,3,4,or 5.