New cubic self-dual codes of length 54, 60 and 66

We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-dual codes with length 54, 60 and 66.Comment: 8 page

On the index system of well-rounded lattices

Let \Lb be a lattice in an $n$-dimensional Euclidean space $E$ and let \Lb' be a Minkowskian sublattice of \Lb, that is, a sublattice having a basis made of representatives for the Minkowski successive minima of \Lb. We consider the set of possible quotients \Lb/\Lb' which may exists in a given dimension or among not too large values of the index [\Lb:\Lb'], indeed [\Lb:\Lb']\le 4, or dimension $n\le 8$.Comment: 17 page