Symplectic groups, symplectic spreads, codes, and unimodular lattices
The symplectic group \(\text{Sp}(2n,p)\) has a complex representation of degree \(p^n\) which splits into two irreducible representations, the Weil representations, of degrees \((p^n-1)/2\) and \((p^n+1)/2\) respectively. One of these representations, say \(W\), has even degree and is faithful. Suppose \(p\equiv 3\pmod 4\) then \(G=\text{Sp}(2n,p)\cdot 2\) has a faithful \(\mathbb{Q}\)-module \(V\) of degree \(p^n+1\) which splits as an \(\text{Sp}(2n,p)\)-module in \(W\) and \(\overline W\) (complex conjugate representation). As the first main result the authors construct in \(V\) a \(G\)-invariant unimodular lattice. In the second main result it is shown that for \(p^n\equiv 3\pmod 4\) this unimodular lattice is unique as a faithful, irreducible, unimodular \(G\)-lattice of degree \(p^n+1\) and that any faithful, irreducible \(G\)-lattice of degree \(p^n+1\) is similar to this lattice or one of two other closely related lattices. The authors handle the case \(p^n\equiv 1\pmod 4\) in a successor of this paper. Finally the following application is given: The construction of the unimodular lattice exhibits an interesting correspondence between selfdual codes of length \(p^3+1\) over \(\text{GF}(p)\) and symplectic spreads in \(\text{GF}(p)^6\). It is well known that for each odd prime \(p\) there are at least two nonisomorphic symplectic spreads in \(\text{GF}(p)^6\) which proves the existence of two selfdual codes of length \(p^3+1\) over \(\text{GF}(p)\). Related work on lattices and Weil representations of \(\text{Sp}(2n,p)\) is [\textit{R. Gow}, J. Algebra 122, No. 2, 510-519 (1989; Zbl 0679.20039)]\ and [\textit{B. H. Gross}, J. Am. Math. Soc. 3, No. 4, 929-960 (1990; Zbl 0745.11035)].
scientific article; zbMATH DE number 1053878
scientific article