The Shadow Theory of Modular and Unimodular Lattices

@article{Rains1998TheST,
  title={The Shadow Theory of Modular and Unimodular Lattices},
  author={Eric M. Rains and N. J. A. Sloane},
  journal={Journal of Number Theory},
  year={1998},
  volume={73},
  pages={359-389}
}
Abstract 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… 

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References

SHOWING 1-10 OF 72 REFERENCES
Nonexistence of Extremal Lattices in Certain Genera of Modular Lattices
Note that (n, l )=(12, 11) is the first open case in the existence problem for extremal _l-modular lattices of dimension n for some similarity _l of norm l with l+1 | 24 (cf. [Que 95] and [ScH 95]).
A new upper bound for the minimum of an integral lattice of determinant 1
Let Λ be an n-dimensional integral lattice of determinant 1. We show that, for all sufficiently large n, the minimal nonzero squared length in Λ does not exceed [ ( n + 6 ) /10 ]. This bound is a
Upper bounds for modular forms, lattices, and codes
Lattices with theta functions for G(√2) and linear codes
A new upper bound on the minimal distance of self-dual codes
It is shown that the minimal distance d of a binary self-dual code of length n>or=74 is at most 2((n+6)/10). This bound is a consequence of some new conditions on the weight enumerator of a self-dual
Low-dimensional lattices. II. Subgroups of GL(n, ℤ)
  • J. ConwayN. Sloane
  • Mathematics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1988
The maximal finite irreducible groups of n x n integers for n = 4, 5, . . . , 9, 11, 13, 17, 19, 23 were determined by Dade, Ryskov, Bülow, Plesken & Pohst and Plesken, as the automorphism groups of
Some elliptic curves arising from the Leech lattice
The theta functions of sublattices of the Leech lattice
Let Λ be the Leech lattice which is an even unimodular lattice with no vectors of squared length 2 in 24-dimensional Euclidean space R 24. Then the Mathieu Group M 24 is a subgroup of the
On Lattices Equivalent To Their Duals
Abstract A lattice is called isodual if it is geometrically congruent to its dual. We show that the densest three-dimensional isodual lattice is the "mean centered-cuboidal" lattice, a lattice which
Shadow Bounds for Self-Dual Codes
  • E. Rains
  • Computer Science
    IEEE Trans. Inf. Theory
  • 1998
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It is shown that a code of length a multiple of 24 meeting the bound cannot be singly-even, and the same technique gives similar results for additive codes over GF(4) (relevant to quantum coding theory).
...
...