The Electronic Journal of Combinatorics

v12i1r22 — Comment by the author, May 31, 2006.

The method used in this paper for constructing perfect 1-factorisations has certainly not been exhausted yet. There are two further orders claimed in reference [10] which I should have recognised in my paper as having been found by Volker Leck. They are 1092728 and 1225044. These can be added to table 6 by adding the lines

p=103, q=1092727, ζ(x)=x3+x+4, c=[344092,1092446]

p=107, q=1225043, ζ(x)=x3+x+9, c=[1107573,151]

Further perfect 1-factorisations can be added to table 6 as follows:

p=139, q=2685619, ζ(x)=x3+x+7, c=[2684547,2435081]

p=151, q=3442951, ζ(x)=x3+x+5, c=[1492322,66]

p=167, q=4657463, ζ(x)=x3+x+3, c=[3183263,109]

p=179, q=5735339, ζ(x)=x3+x+4, c=[4159382,5734770]

Many further results of this type can be found on the author's homepage.

Readers may also care to know the full citation for reference [3], which is

D. Bryant, B. M. Maenhaut and I. M. Wanless, New families of atomic Latin squares and perfect one-factorisations, J. Combin. Theory A 113, (2006) 608-624.