On the Capacity of Constrained Permutation Codes for Rank Modulation

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 lowlevel cells. A permutation s σ Sn satisfies the single-neighbor k-constraint if |δ(i + 1)-δ(i)| = k for all 1 = i = n-1. In this paper, this model is extended into two constraints. A permutation s σ Sn satisfies the two-neighbor k-constraint if for all 2 = i = n-1, |δ(i)-δ(i-1)| = k or |δ(i + 1)-δ(i)| = k, and it satisfies the asymmetric two-neighbor k-constraint if for all 2 = i = n-1, δ(i-1)-δ(i) < k or δ(i + 1)-s(i) < k. We show that the capacity of the first constraint is (1 + e)/2 in case that k = θ(n^e) 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 t-metric.