Upper bounds on the size of quantum codes

Abstract

This paper is concerned with bounds for quantum error-correcting codes. Using the quantum MacWilliams (1972, 1977) identities, we generalize the linear programming approach from classical coding theory to the quantum case. Using this approach, we obtain Singleton-type, Hamming-type, and the first linear-programming-type bounds for quantum codes. Using the special structure of linear quantum codes, we derive an upper bound that is better than both Hamming and the first linear programming bounds on some subinterval of rates.