Etude algorithmique de réseaux construits avec la forme trace. (Algorithmic study of lattices constructed by means of the trace form). (French) Zbl 0787.11024

The authors study the numerical properties of three types of lattices constructed by means of the trace form in cyclotomic number fields and: calculate their minimum and minimal vectors; determine whether or not they are perfect or eutactic. The lattices considered are: certain even unimodular lattices, constructed by E. Bayer-Fluckiger [Comment. Math. Helv. 59, 509-538 (1984; Zbl 0558.10029)], of minimum 4 and dimension 24 (Leech lattice [J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups (1988; Zbl 0634.52002), (2nd ed. 1993; Zbl 0785.11036)]), 32 and 48; certain lattices related to the Leech lattice; and Craig’s lattices (ibid.), constructed using the successive powers of the ideal above \(p\) in the \(p\)-th cyclotomic field.
Reviewer: E.L.Cohen (Ottawa)


11H31 Lattice packing and covering (number-theoretic aspects)
11R18 Cyclotomic extensions
11Y99 Computational number theory


[1] Barnes E. S., Acta Arith. 5 pp 57– (1959)
[2] Bayer-Fluckiger Eva, Comment. Math. Helvetici 59 pp 509– (1984) · Zbl 0558.10029 · doi:10.1007/BF02566364
[3] Bergé A.-M., J. Number Theory 32 pp 14– (1989) · Zbl 0677.10022 · doi:10.1016/0022-314X(89)90095-4
[4] Bergé A.-M., Astérisque 198 pp 41– (1992)
[5] Brauer R., Trans. Roy. Soc. Canada 34 pp 29– (1940)
[6] Conway J. H., Sphere Packings, Lattices and Groups. (1988) · Zbl 0634.52002 · doi:10.1007/978-1-4757-2016-7
[7] Feit W., Proc. London Math. Soc. 29 pp 633– (1974) · Zbl 0312.20003 · doi:10.1112/plms/s3-29.4.633
[8] DOI: 10.1090/S0894-0347-1990-1071117-8 · doi:10.1090/S0894-0347-1990-1071117-8
[9] DOI: 10.1007/BF01457454 · Zbl 0488.12001 · doi:10.1007/BF01457454
[10] DOI: 10.1017/CBO9780511661952 · doi:10.1017/CBO9780511661952
[11] Quebbemann H. G., J. Reine Angew. Math. 326 pp 158– (1981)
[12] Voronoï G., J. Reine Angew. Math. 133 (1908)
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.