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The Shadow Theory of Modular and Unimodular Lattices

Rains, E. M. and Sloane, N. J. A. (1998) The Shadow Theory of Modular and Unimodular Lattices. Journal of Number Theory, 73 (2). pp. 359-389. ISSN 0022-314X. doi:10.1006/jnth.1998.2306.

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It is shown that an n-dimensional unimodular lattice has minimal norm at most 2[n/24]+2, unless n=23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the bound for even unimodular lattices to strongly N-modular even lattices for N in {1, 2, 3, 5, 6, 7, 11, 14, 15, 23}, (*) and analogous bounds are established here for odd lattices satisfying certain technical conditions (which are trivial for N=1 and 2). For N>1 in (*), lattices meeting the new bound are constructed that are analogous to the “shorter” and “odd” Leech lattices. These include an odd associate of the 16-dimensional Barnes–Wall lattice and shorter and odd associates of the Coxeter–Todd lattice. A uniform construction is given for the (even) analogues of the Leech lattice, inspired by the fact that (*) is also the set of square-free orders of elements of the Mathieu group M_(23).

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Additional Information:© 1998 Academic Press. Received January 26, 1998; revised April 22, 1998. The computer language Magma [6], [7], [8] has been helpful in studying particular lattices, testing for modularity, etc.
Issue or Number:2
Record Number:CaltechAUTHORS:20171003-105923947
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Official Citation:E.M Rains, N.J.A Sloane, The Shadow Theory of Modular and Unimodular Lattices, Journal of Number Theory, Volume 73, Issue 2, December 1998, Pages 359-389, ISSN 0022-314X, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81992
Deposited By: Tony Diaz
Deposited On:03 Oct 2017 20:41
Last Modified:15 Nov 2021 19:47

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