Symplectic Groups, Symplectic Spreads, Codes, and Unimodular Lattices Original Research Article

It is known that the symplectic groupSp2n(p) has two (complex conjugate) irreducible representations of degree (pn + 1)/2 realized overimage, provided thatp ≡ 3 mod 4. In the paper we give an explicit construction of an odd unimodularSp2n(p) · 2-invariant lattice Δ(p, n) in dimensionpn + 1 for anypn ≡ 3 mod 4. Such a lattice has been constructed by R. Bacher and B. B. Venkov in the casepn = 27. A second main result says that these lattices are essentially unique. We show that forn ≥ 3 the minimum of Δ(p, n) is at least (p + 1)/2 and at mostp(n − 1)/2. The interrelation between these lattices, symplectic spreads of imagep2n, and self-dual codes over imagepis 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.