Explicit constructions for perfect hash families
Designs, Codes and Cryptography
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A covering array CA(N; t, k, v) is an N × k array, in which in every N × t subarray, each of the vt possible t-tuples over v symbols occurs at least once. The parameter t is the strength of the array. Covering arrays have a wide range of applications, particularly for software interaction testing. Covering arrays are frequently found using computational search. Utilizing a recently developed compact representation of covering arrays known as a covering perfect hash family, in conjunction with the search method known as Tabu search, vastly improved covering arrays for small k are found as well as the first arrays of strength 5, 6, and 7 found by computational search. To build covering arrays of medium strength, the most effective method is to combine small arrays into larger arrays using recursive constructions. One useful type of recursive construction is known as a Roux construction. Roux's original construction was limited to the case where t = 3 and v = 2 and took arrays with k columns and produced arrays with 2k columns. Subsequent generalizations allowed for ℓ k copies, larger v, and t = 4. Further generalizations as well as specializations which greatly improve the size of the resulting array are presented here. A perfect hash family PHF(N; k, v, t) is an N × k array on v symbols with v ≥ t, in which in every N × t subarray, at least one row is comprised of distinct symbols. Perfect hash families also have a wide range of applications, including cryptography, secure frameproof codes, and database management. Perfect hash families have thus far been constructed mainly using direct methods that rely on other combinatorial objects. In addition, several recursive constructions exist. New recursive constructions, new direct constructions, and a large set of PHFs found using Tabu search are provided. These results are then combined with the previous research on perfect hash families to produce tables of the best-known sizes of PHFs for a large set of parameters.