Schulze-Pillot, Rainer Quadratic residue codes and cyclotomic lattices. (English) Zbl 0792.11008 Arch. Math. 60, No. 1, 40-45 (1993). 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. Cited in 2 Documents MSC: 11E12 Quadratic forms over global rings and fields 11H31 Lattice packing and covering (number-theoretic aspects) 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) Keywords:constructions of quadratic forms; cyclotomic fields; quadratic residue codes; Leech lattice; extremal lattice × Cite Format Result Cite Review PDF Full Text: DOI References: [1] W. Feit, On integral representations of finite groups. Proc. London Math. Soc. (3)29, 633-683 (1974). · Zbl 0312.20003 · doi:10.1112/plms/s3-29.4.633 [2] M. Kneser, Klassenzahlen definiter quadratischer Formen. Arch. Math.8, 241-250 (1957). · Zbl 0078.03801 · doi:10.1007/BF01898782 [3] J. Leech andN. J. A. Sloane, Sphere packing and error correcting codes. Canad. J. Math.23, 718-745 (1971). · doi:10.4153/CJM-1971-081-3 [4] F. J.MacWilliams and N. J. A.Sloane, The Theory of Error Correcting Codes, 2nd printing. Amsterdam 1978. · Zbl 0369.94008 [5] O. T.O’Meara, Introduction to Quadratic Forms. Grundlehren d. math. Wiss.117, Berlin-Heidelberg-New York 1973. [6] H. G. Quebbemann, Zur Klassifikation unimodularer Gitter mit Isometrie von Primzahlordnung. J. Reine Angew. Math.326, 158-170 (1981). · Zbl 0452.10027 · doi:10.1515/crll.1981.326.158 [7] B. B. Venkov, Odd unimodular lattices. Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI)86, 40-48 (1979), english transl. J. Soviet Math.17, 1967-1974 (1981). · Zbl 0416.10015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.