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The study of Armstrong codes is motivated by the problem of understandingcomplexities of dependencies in relational database systems, where attributeshave 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 $k-1$ coordinatesthere 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 determinedfor all $q\geq 5$ with three possible exceptions. This disproves a conjectureof Sali. Further, we introduce generalized Armstrong codes for branching, or$(s,t)$-dependencies, construct several classes of optimal Armstrong codes andestablish lower bounds for the maximum length $n$ in this more general setting.

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