Constructions of Snake-in-the-Box Codes under $\ell_{\infty}$-metric for Rank Modulation

In the rank modulation scheme, Gray codes are very useful in the realization of flash memories. For a Gray code in this scheme, two adjacent codewords are obtained by using one "push-to-the-top" operation. Moreover, snake-in-the-box codes under the $\ell_{\infty}$-metric are Gray codes, which can be capable of detecting one $\ell_{\infty}$-error. In this paper, we give two constructions of $\ell_{\infty}$-snakes. On the one hand, inspired by Yehezkeally and Schwartz's construction, we present a new construction of the $\ell_{\infty}$-snake. The length of this $\ell_{\infty}$-snake is longer than the length of the $\ell_{\infty}$-snake constructed by Yehezkeally and Schwartz. On the other hand, we also give another construction of $\ell_{\infty}$-snakes by using $\mathcal{K}$-snakes and obtain the longer $\ell_{\infty}$-snakes than the previously known ones.