On the Capacity of Constrained Permutation Codes for Rank Modulation

Abstract

Motivated by the rank modulation scheme, a recent study by Sala and Dolecek explored the idea of constraint codes for permutations. The constraint studied by them is inherited by the inter-cell interference phenomenon in flash memories, where high-level cells can inadvertently increase the level of low-level cells. A permutation $\sigma \in S_{n}$ satisfies the single-neighbor $k$ -constraint if $|\sigma (i+1)-\sigma (i)|\leq k$ for all $1\leq i\leq n-1$ . In this paper, this model is extended into two constraints. A permutation $\sigma \in S_{n}$ satisfies the two-neighbor $k$ -constraint if for all $2 \leq i \leq n-1$ , $|\sigma (i)-\sigma (i-1)|\leq k$ or $|\sigma (i \,\, + \,\, 1)-\sigma (i)|\leq k$ , and it satisfies the asymmetric two-neighbor $k$ -constraint if for all $2 \leq i \leq n-1$ , $\sigma (i-1)-\sigma (i) or $\sigma (i+1)-\sigma (i) . We show that the capacity of the first constraint is $(1 \,\, + \,\, \epsilon )/2$ in case that $k=\Theta (n^{\epsilon })$ and the capacity of the second constraint is 1 regardless for any positive $k$ . We also extend our results and study the capacity of these two constraints combined with error-correcting codes in the Kendall $\tau $ -metric.