Quantum shadow enumerators

In a previous paper, Shor and Laflamme (see Phys. Rev. Lett., vol.78, p.1600-02, 1997) define two "weight enumerators" for quantum error-correcting codes, connected by a MacWilliams (1977) 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) can be shown to be optimal among all quantum codes. We also use the shadow machinery to extend a bound on additive codes to general codes, obtaining as a consequence that any code of length, can correct at most [(n+1)/6] errors