We present a systematic way to construct p-ary quantum error correcting codes using logic functions. As a consequence, for a given function with APC distance d′ 2, we can construct quantum codes with parameters ((n, K, d))p and gain a lower bound of K for all 2 d d′. The basic states of the constructed quantum codes can be stated and the sufficient conditions for saturating quantum Singleton bound are also discussed. We give quantum codes [[5, 1, 3]]p with p prime, [[6, 0, 4]], [[6, 2, 3]]p with p > 2 prime and [[2n, 2n - 2, 2]] as examples constructed in this way. As a consequence, for a given function with APC distance d ’2, we can construct quantum codes with parameters ((n, K, d)) p and gain a lower bound of K for all 2 dd ’. The basic states of the constructed quantum codes can be stated and the sufficient conditions for saturating quantum Singleton bound are also discussed. We give quantum codes [[5, 1, 3]] p with p prime, [[6, 0, 4]], [[6, 2, 3]] p with 2 prime and [[2n, 2n - 2, 2]] as examples constructed in this way.