Let π n be a uniformly chosen random permutation on [n]. Using an analysis of the probability that two overlapping consecutive kpermutations are order isomorphic, we show that the expected number of distinct consecutive patterns in π n is n 2 2 (1 − o( 1)). This exhibits the fact that random permutations pack consecutive patterns nearperfectly.

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In this article we improve the dimension and minimum distance bound of the the Hermitian Lifted Codes LRCs construction from López, Malmskog, Matthews, Piñero and Wooters (López et. al.) via elementary univariarte polynomial division. They gave an asymptotic rate estimate of 0.007. We improve the rate estimate to 0.1 using univariate polynomial division