The existence of Room 5-cubes
A Room t-cube of side n is a t-dimensional array of side n on a set S of \(n+1\) symbols such that: (i) Each cell is either empty or contains an unordered pair of distinct symbols from S, (ii) Each symbol occurs in every (t-1)-dimensional flat in the array exactly once, (iii) Every unordered pair of symbols occurs precisely once in the array. It is known that the existence of a Room t-cube of side n is equivalent to the existence of t pairwise orthogonal symmetric Latin squares of side n. In this paper the author proves that if \(n\geq 11\) is odd (except possibly \(n=15)\), then there is a Room 5-cube of side n. The proof involves a recursive construction using 5-dimensional frames. The author also conjectures that there is a Room 5-cube of side 15.
scientific article; zbMATH DE number 4039920
scientific article