SYMPLECTIC GROUPS, SYMPLECTIC SPREADS, CODES, AND UNIMODULAR LATTICES

It is known that the symplectic group Sp(2n)(p) has two (complex conju gate) irreducible representations of degree (p(n) + 1)/2 realized over Q(root - p), provided that p = 3 mod 4. In the paper we give an expli cit construction of an odd unimodular Sp(2n)(p).2-invariant lattice De lta(p, n) in dimension p(n) + 1 for any p(n) = 3 mod 4. Such a lattice has been constructed by R. Bacher and B. B. Venkov in the case p(n) = 27. A second main result says that these lattices are essentially uni que. We show that for n greater than or equal to 3 the minimum of Delt a(p, n) is at least (p + 1)/2 and at most p((n-1)/2). The interrelatio n between these lattices, symplectic spreads of F-p(2n), and self-dual codes over F-p is also investigated. In particular, using new results of U. Dempwolff and L. Bader, W. M. Kantor, and G. Lunardon, we come to three extremal self-dual ternary codes of length 28. (C) 1997 Acade mic Press.