Snake-in-the-Box Codes for Rank Modulation under Kendall's $\tau$-Metric

TL;DR

This paper develops new snake-in-the-box codes under Kendall's tau-metric for rank modulation in flash memories, providing constructions with larger sizes and solving open problems in the field.

Contribution

It proves the validity of a known construction and introduces a new, larger snake-in-the-box code construction for permutations under Kendall's tau-metric.

Findings

Validated a previous snake construction with size (2n+1)!/2 - 2n + 1

Proposed a new construction achieving size (2n+1)!/2 - 2n + 3

Successfully applied the new construction to S7

Abstract

For a Gray code in the scheme of rank modulation for flash memories, the codewords are permutations and two consecutive codewords are obtained using a push-to-the-top operation. We consider snake-in-the-box codes under Kendall's $\tau$-metric, which is a Gray code capable of detecting one Kendall's $\tau$-error. We answer two open problems posed by Horovitz and Etzion. Firstly, we prove the validity of a construction given by them, resulting in a snake of size $M_{2n+1}=\frac{(2n+1)!}{2}-2n+1$. Secondly, we come up with a different construction aiming at a longer snake of size $M_{2n+1}=\frac{(2n+1)!}{2}-2n+3$. The construction is applied successfully to $S_7$.