Abstract-New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths. Second, several probabilistic lower bounds are developed, which improve on the known lower bounds for large minimum distances. The results of a computer search for permutation codes are also presented.

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a b s t r a c tWe study F -saturation games, first introduced by Füredi, Reimer and Seress in 1991, and named as such by West (2009). The main question is to determine the length of the game whilst avoiding various classes of graph, playing on a large complete graph. We show lower bounds on the length of path-avoiding games, and more precise results for short paths. We show sharp results for the tree avoiding game and the star avoiding game.

We propose a new model of asynchronous batch codes that allow for parallel recovery of information symbols from a coded database in an asynchronous manner, i.e. when different queries take different time to process. Then, we show that the graph-based batch codes studied by Rawat et al. are asynchronous. Further, we demonstrate that hypergraphs of Berge girth at least 4, respectively at least 3, yield graphbased asynchronous batch codes, respectively private information retrieval (PIR) codes. We prove the hypergraph-theoretic proposition that the maximum number of hyperedges in a hypergraph of a fixed Berge girth equals the quantity in a certain generalization of the hypergraph-theoretic (6,3)-problem, first posed by Brown, Erdős and Sós. We then apply the constructions and bounds by Erdős, Frankl and Rödl about this generalization of the (6,3)problem, known as the (3r-3,r)-problem, to obtain batch code constructions and bounds on the redundancy of the graph-based asynchronous batch and PIR codes. Finally, we show that the optimal redundancy ρ(k) of graph-based asynchronous batch codes of dimension k with the query size t = 3 is 2 √ k. Moreover, for a general fixed value of t ≥ 4, ρ(k) = O k 1/(2−ǫ) for any small ǫ > 0. For a general value of t ≥ 4, lim k→∞ ρ(k)/ √ k = ∞. Index Terms-primitive linear multiset batch codes, private information retrieval codes, extremal hypergraph theory, Turán theory, packing designs.

We study analogues of F-saturation games, first introduced by Füredi, Reimer and Seress [2] in 1991, and named as such by West [8]. We examine analogous games on directed graphs, and show tight results on the walk-avoiding game. We also examine an intermediate game played on undirected graphs, such that there exists an orientation avoiding a given family of directed graphs, and show bounds on the score. This last game is shown to be equivalent to a recent game studied by Hefetz, Krivelevich, Naor, and Stojakovićin [5], and we give new bounds for biased versions of this game.