Quadratic residue codes and cyclotomic lattices. (English) Zbl 0792.11008

Summary: Constructions of unimodular positive definite quadratic forms via quadratic residue codes over finite fields and via ideals in cyclotomic number fields are compared in this article. In particular one obtains a natural generalization of the coding theoretic constructions of the Leech lattice and of the extremal lattice \(P48q\) in dimension 48 to higher dimensions. The lattices obtained could be extremal in dimensions 72 and 80, but at present the author sees no way to prove this.


11E12 Quadratic forms over global rings and fields
11H31 Lattice packing and covering (number-theoretic aspects)
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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