On the snake-in-the-box codes for rank modulation under Kendall’s $$\tau $$ τ -metric

Abstract

For exactly and efficiently representing and storing data in flash memories, the rank modulation scheme has been presented. In this scheme, Gray codes over the permutations are important, which are used to represent information in flash memories. For a Gray code, two consecutive codewords are obtained using one “push-to-the-top” operation. Specially, a snake-in-the-box code under the Kendall’s \(\tau \)-metric is a Gray code, which is capable of detecting one Kendall’s \(\tau \)-error. In this paper, we consider only the Kendall’s \(\tau \)-metric on the permutations. And we answer one open problem proposed by Horovitz and Etzion. That is, we prove that the length of the longest snake in \(S_{2n+2}\) is longer than the length of the longest snake in \(S_{2n+1}\).