Complexity of Dependencies in Bounded Domains, Armstrong Codes, and Generalizations

06/14/2019

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by   Yeow Meng Chee, et al.

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The study of Armstrong codes is motivated by the problem of understanding complexities of dependencies in relational database systems, where attributes have bounded domains. A (q,k,n)-Armstrong code is a q-ary code of length n with minimum Hamming distance n-k+1, and for any set of k-1 coordinates there exist two codewords that agree exactly there. Let f(q,k) be the maximum n for which such a code exists. In this paper, f(q,3)=3q-1 is determined for all q≥ 5 with three possible exceptions. This disproves a conjecture of Sali. Further, we introduce generalized Armstrong codes for branching, or (s,t)-dependencies, construct several classes of optimal Armstrong codes and establish lower bounds for the maximum length n in this more general setting.