A Kirkman square of order 51 and block size 3

Abstract

A necessary condition for the existence of a Kirkman square KS3(v) is v ≡ 3 (mod 6). It is known that for all v sufficiently large and v ≡ 3 (mod 6) that such a design exists. In order to settle the existence question completely a number of small designs must be constructed directly. Of the 14 possible orders less than 100 only 4 are currently known to exist. In this paper we establish the existence of one more design with order less than 100; namely, v = 51.