, which also provided a general lower bound on the maximal achievable minimum distance dmax(n, k, r, δ) that a linear LRC with parameters (n, k, r, δ) can have. This article expands the class of parameters (n, k, d, r, δ) for which there exist perfect linear LRCs and improves the lower bound for dmax(n, k, r, δ). Further, this bound is proved to be optimal for the class of matroids that is used to derive the existence bounds of linear LRCs.
Clustering data in Euclidean space has a long tradition and there has been considerable attention on analyzing several different cost functions. Unfortunately these result rarely generalize to clustering of categorical attribute data. Instead, a simple heuristic k-modes is the most commonly used method despite its modest performance. In this study, we model clusters by their empirical distributions and use expected entropy as the objective function. A novel clustering algorithm is designed based on local search for this objective function and compared against six existing algorithms on well known data sets. The proposed method provides better clustering quality than the other iterative methods at the cost of higher time complexity.
Consider an n × n × n cube Q consisting of n 3 unit cubes. A tripod of order n is obtained by taking the 3n − 2 unit cubes along three mutually adjacent edges of Q. The unit cube corresponding to the vertex of Q where the edges meet is called the center cube of the tripod. The function f (n) is defined as the largest number of integral translates of such a tripod that have disjoint interiors and whose center cubes coincide with unit cubes of Q. The value of f (n) has earlier been determined for n ≤ 9. The function f (n) is here studied in the framework of the maximum clique problem, and the values f (10) = 32 and f (11) = 38 are obtained computationally. Moreover, by prescribing symmetries, constructive lower bounds on f (n) are obtained for n ≤ 26. A conjecture that f (n) is always attained by a packing with a symmetry of order 3 that rotates Q around the axis through two opposite vertices is disproved.
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