### Abstract

It is known that the symplectic group Sp_{2n}(p) has two (complex conjugate) irreducible representations of degree (p^{n} + 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 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 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 sign^{2n}_{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.

Original language | English (US) |
---|---|

Pages (from-to) | 113-156 |

Number of pages | 44 |

Journal | Journal of Algebra |

Volume | 194 |

Issue number | 1 |

State | Published - Aug 1 1997 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*194*(1), 113-156.

**Symplectic groups, symplectic spreads, codes, and unimodular lattices.** / Scharlau, Rudolf; Tiep, Pham Huu.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 194, no. 1, pp. 113-156.

}

TY - JOUR

T1 - Symplectic groups, symplectic spreads, codes, and unimodular lattices

AU - Scharlau, Rudolf

AU - Tiep, Pham Huu

PY - 1997/8/1

Y1 - 1997/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0031206584&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031206584&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031206584

VL - 194

SP - 113

EP - 156

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -