New lattice sphere packings denser than Mordell-Weil lattices

1) We present new lattice sphere packings in Euclid spaces of many dimensions in the range 3332-4096, which are denser than known densest Mrodell-Weil lattice sphere packings in these dimensions. Moreover it is proved that if there were some nice linear binary codes we could construct lattices denser than Mordell-Weil lattices of many dimensions in the range 116-3332. 2) New lattices with densities at least 8 times of the densities of Craig lattices in the dimensions $p-1$, where $p$ is a prime satisfying $p-1 \geq 1222$, are constructed. Some of these lattices provide new record sphere packings. 3) Lattice sphere packings in many dimensions in the range 4098-8232 better than present records are presented. Some new dense lattice sphere packings in moderate dimensions $84, 85, 86, 181-189$ denser than any previously known sphere packings in these dimensions are also given. The construction is based on the analogues of Craig lattices.