Constrained Codes for Rank Modulation

Motivated by the rank modulation scheme, a recent work by Sala and Dolecek explored the study 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. In this paper, the model studied by Sala and Dolecek is extended into two constraints. A permutation $σ\in S_n$ satisfies the \emph{two-neighbor $k$-constraint} if for all $2 \leq i \leq n-1$ either $|σ(i-1)-σ(i)|\leq k$ or $|σ(i)-σ(i+1)|\leq k$, and it satisfies the \emph{asymmetric two-neighbor $k$-constraint} if for all $2 \leq i \leq n-1$, either $σ(i-1)-σ(i) < k$ or $σ(i+1)-σ(i) < k$. We show that the capacity of the first constraint is $(1+ε)/2$ in case that $k=Θ(n^ε)$ and the capacity of the second constraint is 1 regardless to the value of $k$. We also extend our results and study the capacity of these two constraints combined with error-correction codes in the Kendall's $τ$ metric.

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