On maximal partial Latin hypercubes

Donovan, Diane. M.; Grannell, Mike J. and Yazici, Emine. Sule (2024). On maximal partial Latin hypercubes. Designs, Codes and Cryptography, 92(2) pp. 419–433.

DOI: https://doi.org/10.1007/s10623-023-01314-5

Abstract

A lower bound is presented for the minimal number of filled cells in a maximal partial Latin hypercube of dimension d and order n. The result generalises and extends previous results for d = 2 (Latin squares) and d = 3 (Latin cubes). Explicit constructions show that this bound is near-optimal for large n > d. For d > n, a connection with Hamming codes shows that this lower bound gives a related upper bound for the same quantity. The results can be interpreted in terms of independent dominating sets in certain graphs, and in terms of codes that have covering radius 1 and minimum distance at least 2.

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