The number of filled cells in a maximal partial latin hypercube

This thesis provides a comprehensive literature review on the minimum number of filled cells, f(d,n), in a Maximal Partial Latin Hypercube (MPLH) of order n and dimension d. Latin hypercubes generalize the concept of Latin squares and cubes to higher dimensions. The review examines significant findings on lower bounds for f(d,n), beginning with smaller cases where d=2 and d=3. The study also explores results on the spectrum of Maximal Partial Latin squares and cubes. Additionally, interdisciplinary approaches involving graph theory and coding theory are discussed, emphasizing their contributions to the understanding of MPLHs. This work serves as a foundation for future research, providing a thorough overview of the methodologies and findings that have shaped our current knowledge of MPLHs.