In a recent paper by the same author general improvements on the sphere covering bound for binary covering codes were obtained. In the present work it is shown how the main idea can be exploited to obtain "improved sphere bounds" also for nonbinary codes. We concentrate on generalq-ary codes and on binary/ternary mixed codes. Special attention is paid to the football pool problem; a few new lower bounds are established. Also, it is shown how a similar method yields upper bounds on packing codes.