191 research outputs found
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
Sphere Packings in Euclidean Space with Forbidden Distances
In this paper, we study the sphere packing problem in Euclidean space, where
we impose additional constraints on the separations of the center points. We
prove that any sphere packing in dimension , with spheres of radii ,
such that \emph{no} two centers and satisfy , has density less or equal than . Equality occurs if and only if the packing is given by a
-dimensional even unimodular extremal lattice. This shows that any of the
lattices and are optimal for this
constrained packing problem. We also give results for packings up to dimension
, where we impose constraints on the distance between centers and
on the minimal norm of the spectrum, and show that even unimodular extremal
lattices are again uniquely optimal. Moreover, in the -dimensional case, we
give a condition on the set of constraints that allow the existence of an
optimal periodic packing, and we develop an algorithm to find them by relating
the problem to a question about linear domino tilings.Comment: 39 page
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