  In this paper, based on the relationship between logic functions and quantum
error correcting codes(QECCs), we unify the construction of QECCs via graphs,
projectors and logic functions. A construction of QECCs over a prime field
GF(p) is given, and one of the results given by Ref[8] can be viewed as a
corollary of one theorem in this paper. With the help of Boolean functions, we
give a clear proof of the existence of a graphical QECC in mathematical view,
and find that the existence of an [[n,k,d]] QECC over GF(p) requires similar
conditions with that depicted in Ref[9]. The result that under the
correspondence defined in Ref[17], every [[n,0,d]] QECC over GF(2)
corresponding to a simple undirected graph has a Boolean basis state, which is
closely related to the adjacency matrix of the graph, is given.
  After a modification of the definition of operators, we find that some QECCs
constructed via projectors depicted in Ref[11] can have Boolean basis states. A
necessary condition for a Boolean function being used in the construction via
projectors is given. We also give some examples to illustrate our results.
