We prove that the kissing numbers in 17, 18, 19, 20, and 21 dimensions are at least 5730, 7654, 11692, 19448, and 29768, respectively. The previous records were set by Leech in 1967, and we improve on them by 384, 256, 1024, 2048, and 2048. Unlike the previous constructions, the new configurations are not cross sections of the Leech lattice minimal vectors. Instead, they are constructed by modifying the signs in the lattice vectors to open up more space for additional spheres.