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An Algorithm for Enumerating Distinct Cyclic Steiner Systems

Timothy C. Frenz1, Donald L. Kreher2
1School of Computer and Information Science Center for Science and Technology Syracuse University Syracuse, NY 13244-4100 U.S.A.
2Department of Mathematical Science Michigan Technological University Houghton, Michigan 49931 USS.A.

Abstract

An algorithm is presented for finding all \((0,1)\)-solutions to the matrix problem \(AX = J\), where \(A\) is a \((0,1)\)-matrix and \(J\) is the all \(1\)’s column vector. It is applied to the problem of enumerating distinct cyclic Steiner systems and five new values are obtained. Specifically, the number of distinct solutions to \(S(2,3,55), S(2,3,57), S(2,3,61), S(2,3,63)\), and \(S(3,4,22)\) are \(121,098,240, 84,672,512, 2,542,203,904, 1,782,918,144\), and \(1140\), respectively.