Quasi-symmetric designs on $ 56 $ points

Abstract

Computational techniques for the construction of quasi-symmetric block designs are explored and applied to the case with 56 points. One new (56,16,18) and many new (56, 16, 6) designs are discovered, and non-existence of (56, 12, 9) and (56, 20, 19) designs with certain automorphism groups is proved. The number of known symmetric (78,22,6) designs is also significantly increased.

Cubes of symmetric designs

“…The last column of Table 2 (Nng) contains their distribution by slice invariants. The constructed examples are available on our web page [23]. The total number of non-group cubes in C 3 (16, 6, 2) is probably much larger.…”

“…The parameters for which we found non-difference group cubes are given in Table 3. An on-line version of the table with links to files containing the cubes is available on the web page [23]. The column Nds contains numbers of inequivalent difference sets according to [30].…”

Strongly regular configurations

“…The construction method for configurations with prescribed automorphism groups is similar to constructions of quasi-symmetric designs in [32,30] and relies on the clique-finding program cliquer [37]. Another family of semipartial geometries is family (g) from [14], denoted by LP (n, q) in [19,15].…”

A new partial geometry pg(5,5,2)

“…In this note we construct another partial geometry with parameters pg(5, 5, 2), not isomorphic to the partial geometry of van Lint and Schrijver. The new partial geometry was discovered by prescribing automorphism groups and preforming computer searches, with techniques similar to the ones used in [7] for quasi-symmetric designs. In Section 2 we describe a construction of the new pg(5, 5, 2) by changing some lines of the geometry of van Lint and Schrijver.…”