Quantum error detection

We show that the probability of undetected error for a quantum code on the depolarizing channel can be expressed via code´s weight enumerators. We prove that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive a lower bound on this exponent. To derive upper bounds we formulate a linear programming problem and present two feasible programs for it. The asymptotic upper and lower bounds coincide in a certain interval of code rates close to 1