  In a recent paper [quant-ph/9610040], Shor and Laflamme define two ``weight
enumerators'' for quantum error correcting codes, connected by a MacWilliams
transform, and use them to give a linear-programming bound for quantum codes.
We extend their work by introducing another enumerator, based on the classical
theory of shadow codes, that tightens their bounds significantly. In
particular, nearly all of the codes known to be optimal among additive quantum
codes (codes derived from orthogonal geometry ([quant-ph/9608006])) can be
shown to be optimal among all quantum codes. We also use the shadow machinery
to extend a bound on additive codes (E. M. Rains, manuscript in preparation) to
general codes, obtaining as a consequence that any code of length n can correct
at most floor((n+1)/6) errors.
