Symplectic groups, symplectic spreads, codes, and unimodular lattices

It is known that the symplectic group Sp_2n(p) has two (complex conjugate) irreducible representations of degree (pⁿ + 1)/2 realized over ℚ(√-p), provided that p ≡ 3 mod 4. In the paper we give an explicit construction of an odd unimodular Sp_2n(p) · 2-invariant lattice Δ(p, n) in dimension pⁿ + 1 for any pⁿ ≡ 3 mod 4. Such a lattice has been constructed by R. Bacher and B. B. Venkov in the case pⁿ = 27. A second main result says that these lattices are essentially unique. We show that for n ≥ 3 the minimum of Δ(p, n) is at least (p + 1)/2 and at most p^{(n - 1)/2}. The interrelation between these lattices, symplectic spreads of double-struck F sign²ⁿ_p, and self-dual codes over double-struck F sign_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.