Some \(6\times 11\) Youden squares and double Youden rectangles
A Youden square of size \(k\times v\) is a rectangular (not square) arrangement of \(kv\) entries \(x\) in \(k\) rows and \(v\) columns, \(k<v\), such that (i) each entry \(x\) is drawn from a set \(S1\) of \(v\) elements; (ii) each element from \(S1\) occurs exactly once in each row and no more than once per column; (iii) each pair from \(S1\) occur together in exactly \(\lambda\) columns, where \(\lambda =k(k-1)/(v-1)\), i.e. the sets of elements of \(S1\) in the columns are the blocks of a symmetric balanced incomplete block design (SBIBD). A double Youden rectangle (DYR) of size \(k\times v\) is an arrangement of the \(kv\) pairs \(x,y\) in \(k\) rows and \(v\) columns, \(k<v\), such that (i) each value \(x\) is drawn from a set \(S1\) of \(v\) elements; (ii) each value \(y\) is drawn from a set \(S2\) of \(k\) elements; (iii) each element from \(S1\) occurs exactly once in each row and no more than once per column; (iv) each element from \(S2\) occurs exactly once in each column and either \(n\) or \(n+1\) times in each row, where \(n\) is the integral part of \(v/k\); (v) each element from \(S1\) is paired exaclty once with each element from \(S2\); (vi) the sets of elements of \(S1\) in the columns are the blocks of an SBIBD with parameters \(\{v,k,\lambda\}\); (vii) if \(n\) occurrences of each element from \(S2\) are removed from each row, the remaining sets of elements of \(S2\) in the rows are the blocks of an SBIBD, or else just a single element from \(S2\) remains in each row. In this paper, 5-cyclic Youden squares of size \(6\times 11\) are enumerated and classified into 70 species. The author also investigates the existence of 5-cyclic \(6 \times 11\) double Youden rectangles and constructs a number of new designs of this type.
symmetric balanced incomplete block design
scientific article; zbMATH DE number 1018540