Asymptotic and constructive methods for covering perfect hash families and covering arrays

• Published in: Designs, codes, and cryptography Vol. 86; no. 4; pp. 907 - 937
• Main Authors: Colbourn, Charles J., Lanus, Erin, Sarkar, Kaushik
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2018; Springer Nature B.V
• Subjects: Algorithms; Arrays; Asymptotic methods; Circuits; Coding and Information Theory; Columns (structural); Computer Science; Construction methods; Cryptology; Data Structures and Information Theory; Discrete Mathematics in Computer Science; Hash based algorithms; Information and Communication; Polynomials; Probabilistic methods; Resampling; Strength; Upper bounds; Conditional expectation algorithm; Asymptotic bound; 05B40; 51E20; 05D40; 05E18; Covering perfect hash family; Covering array; 05B15
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-017-0369-x

Covering perfect hash families represent certain covering arrays compactly. Applying two probabilistic methods to covering perfect hash families improves upon the asymptotic upper bound for the minimum number of rows in a covering array with v symbols, k columns, and strength t . One bound can be realized by a randomized polynomial time construction algorithm using column resampling, while the other can be met by a deterministic polynomial time conditional expectation algorithm. Computational results are developed for both techniques. Further, a random extension algorithm further improves on the best known sizes for covering arrays in practice. An extensive set of computations with column resampling and random extension yields explicit constructions when k ≤ 75 for strength seven, k ≤ 200 for strength six, k ≤ 600 for strength five, and k ≤ 2500 for strength four. When v > 3 , almost all known explicit constructions are improved upon. For strength t = 3 , restrictions on the covering perfect hash family ensure the presence of redundant rows in the covering array, which can be removed. Using restrictions and random extension, computations for t = 3 and k ≤ 10 , 000 again improve upon known explicit constructions in the majority of cases. Computations for strengths three and four demonstrate that a conditional expectation algorithm can produce further improvements at the expense of a larger time and storage investment.