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 l∞-metric are Gray codes, which can be capable of detecting one l∞-error. In this paper, we give two constructions of l∞snakes. On the one hand, inspired by Yehezkeally and Schwartz’s construction, we present a new construction of the l∞-snake. The length of this l∞-snake is longer than the length of the l∞-snake constructed by Yehezkeally and Schwartz. On the other hand, we also give another construction of l∞-snakes by using K-snakes and obtain the longer l∞-snakes than the previously known ones.