Codes of Nets with Translations

| ij ∩ i′j′ | =   1 if i = i′; 0 if i= i′, j = j′; n if i= i′, j= j′. Thus the lines are partitioned into k parallel classes { ij : 1≤ j≤n}, 1≤ i≤ k, and each line has n points. Note that an (n+1)-net is the same thing as an affine plane of order n. For any k-net N and any k′-subset of the parallel classes of N , we obtain a k′-subnet N ′ ⊆ N of order n, also with point set P. The description of nets found here is more amenable to our specialized rank computations than to a broader presentation; more general information concerning nets is found in Baer (1939), Beth et al. (1985), Bruck (1951) and (1963), Jungnickel (1990), and Pickert (1975). We denote the p-rank of a net N (the p-rank of its incidence matrix) by rankp N . In Moorhouse (1990) we proposed the following conjecture, supported by numerous computational examples: