Nets and Latin Squares of Order 11

As part of my list of nets of small orders, I am compiling here a list of nets of order 11. This list is by no means complete. So far I have included all nets formed by orthogonal mates of the cyclic Latin square of order 11, for which the resulting 4-net contains no affine plane of order 2; and all subnets of these.

Update Friday Feb 10, 2006:   I have just determined that the only 4-nets N of order 11 having the 4-tuple (X,X,X,X) in V₀(N) (in the notation of my manuscripts Ranks of Nets and Ranks of Nets and Webs) are those listed as 4₂, 4₃, 4₄, 4₅, 4₆, 4₇ and 4₁₀ below. This result used weeks of computational time on our department's Beowulf cluster acquired under the National Science Foundation under Grant No. 0421935.

In addition to my own C++ programs, I have made use of Brendan McKay's graph isomorphism package nauty.

If you are aware of errors or omissions in my list, I would appreciate an email message () from you.

Each line of the following table denotes a distinct isomorphism class of k-net of order 11. In each case I have included

• A name for the net consists of the integer k (the number of parallel classes), together with a subscript distinguishing the isomorphism class among all k-nets of order 11. For k=3, a link is provided to a file containing the lexicographically smallest Latin square representing the 3-net. For k≥4, a link is provided to a set of k−2 MOLS (mutually orthogonal Latin squares of order 11) representing the net; in this case the choice of canonical representative is more complicated to describe.
• The order of automorphism group G of the net.
• The order of the subgroup H≤G consisting of all parallel-class preserving automorphisms of the net.
• A list of isomorphism types of the (k−1)-subnets of the k-net, followed (in parentheses) by a number indicating how many of the k subnets have this type.
• A list of all possible extensions to (k+1)-nets.
• Rows highlighted in pink indicate the Desarguesian nets; these are all the subnets of the Desarguesian affine plane of order 11.

/ revised February, 2006