Snake-in-the-Box Codes for Rank Modulation under Kendall’s $\tau $ -Metric in $S_{2n+2}$

Snake-in-the-box codes under Kendall’s
$\tau $
-metric are studied in the rank modulation scheme for flash memories, where codewords are a subset of permutations in
$S_{n}$
with minimal Kendall’s
$\tau $
-distance two, and two cyclically consecutive codewords are connected via a push-to-the-top operation. Studies so far restrict the push-to-the-top operations only on odd indices, resulting in a snake consisting of permutations with the same parity, and thus, the minimal distance constraint is easily satisfied. Asymptotically optimal snake codes have been constructed this way in
$S_{2n+1}$
. As for
$S_{2n+2}$
, this framework keeps the last element fixed, and thus, a snake in
$S_{2n+2}$
is equivalent to a snake in
$S_{2n+1}$
, which is rather trivial. If one wants to do better, then it is inevitable to have some push-to-the-top operations on even indices, resulting in a combination of odd and even permutations in the snake, which increases the difficulty to guarantee the minimal Kendall’s
$\tau $
-distance constraint. Thus, Horovitz and Etzion pose the open problem to prove or disprove that the size of the largest snake in
$S_{2n+2}$
is not larger than the size of the largest snake in
$S_{2n+1}$
. A first step toward this problem is a negative answer by Wang and Fu, who construct a snake in
$S_{2n+2}$
with exactly one more permutation than an optimal snake in
$S_{2n+1}$
. In this paper, we give an explicit construction of a snake in
$S_{2n+2}$
with size asymptotically approaching
$({1}/{4})|S_{2n+2}|$
.