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In the rank modulation scheme, Gray codes are very useful in the realizationof flash memories. For a Gray code in this scheme, two adjacent codewords areobtained by using one "push-to-the-top" operation. Moreover, snake-in-the-boxcodes under the $\ell_{\infty}$-metric are Gray codes, which can be capable ofdetecting one $\ell_{\infty}$-error. In this paper, we give two constructionsof $\ell_{\infty}$-snakes. On the one hand, inspired by Yehezkeally andSchwartz's construction, we present a new construction of the$\ell_{\infty}$-snake. The length of this $\ell_{\infty}$-snake is longer thanthe length of the $\ell_{\infty}$-snake constructed by Yehezkeally andSchwartz. 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.

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