03973181

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03973181 B\'en\'eteau, Lucien Lacaze, Jacqueline Designs arising from symplectic geometry. Applied algebra, algorithmics and error-correcting codes, Proc. 2nd Int. Conf., Toulouse/France 1984, Lect. Notes Comput. Sci. 228, 198-205 (1986). 1986 EN affine Steiner triple systems commutative Moufang loop Hall triple systems Zbl 0591.00035 Zbl 0482.20044 Zbl 0573.20065 Zbl 0527.05018 Zbl 0373.20062 Zbl 0503.05008 [For the entire collection see Zbl 0591.00035.] The affine Steiner triple systems arise from exponent 3 abelian 3-groups $(A,+)$ by defining the blocks as the 3-element subsets of the form $\{x,2(x+y),y\}$. If more generally $(A,+)$ is assumed to be an exponent 3 commutative Moufang loop, one obtains some $2$-(3${}\sp s,3,1)$ designs that are generally called Hall triple systems (HTSs). When $s<8$ these systems are known to have corresponding commutative Moufang loops of class $<3$ [see {\it L. B\'en\'eteau}, Ann. Fac. Sci. Toulouse, V. S\'er., Math. 3, 75-88 (1981; Zbl 0482.20044)]; therefore they can be given simple exterior algebra constructions (see for instance [{\it T. Kepka} and {\it P. Neme\v{c}}, Czech. Math. J. 31(106), 633-669 (1981; Zbl 0573.20065)], [{\it R. Roth}, {\it D. K. Ray-Chaudhuri}, J. Comb. Theory, Ser. A 36, 129-162 (1984; Zbl 0527.05018)]. The previously known classification results about "small" HTSs concern the situations $s<7$, and $r<6$ where r is the rank (cardinal number of minimal generator sets); c.f. [{\it T. Kepka}, Commentat. Math. Univ. Carol. 19, 389-401 (1978; Zbl 0373.20062)] and [{\it L. B\'en\'eteau}, Ann. Discrete Math. 18, 55-60 (1983; Zbl 0503.05008)]. The paper under review yields the classification of the Hall triple systems whose 3-order and rank are both 7. The fact that these designs form exactly 5 isomorphism classes is obtained as a by-product of some results concerning symplectic trilinear forms of the vector space V(n,K), where K is an arbitrary field of characteristic $\ne 2$. The set of these forms is provided with a natural action of the linear group. The problem of determining the orbits depends on the field as soon as $n>5$. When $K=GF(3)$ this problem is shown to be equivalent to cataloging the HTSs satisfying $s=(n+1)=r$. The special case of the HTSs admitting a codimension 1 affine subsystem is shown to consist of [(n-1)/2] isomorphism classes. Recall that the cubic hypersurface quasigroups (introduced by Manin) and the centerless special Fischer groups are also known to provide algebraic descriptions of these designs.