Symplectic groups, symplectic spreads, codes, and unimodular lattices

Rudolf Scharlau, Pham Huu Tiep

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


It is known that the symplectic group Sp2n(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 Sp2n(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 sign2np, and self-dual codes over double-struck F signp 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.

Original languageEnglish (US)
Pages (from-to)113-156
Number of pages44
JournalJournal of Algebra
Issue number1
StatePublished - Aug 1 1997

ASJC Scopus subject areas

  • Algebra and Number Theory


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