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

## TL;DR

This paper investigates the maximum lengths of Armstrong codes related to dependency complexities in bounded domain databases, proving a key formula, disproving a conjecture, and introducing generalized codes with new bounds.

## Contribution

It determines the maximum length for certain Armstrong codes, disproves a prior conjecture, and introduces generalized Armstrong codes with new bounds.

## Key findings

- Exact formula for $f(q,3)$ for all $q \\geq 5$ with three exceptions
- Disproof of Sali's conjecture on Armstrong codes
- Construction of new classes of optimal generalized Armstrong codes

## Abstract

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 $q\geq 5$ 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.