  The challenge of quantum computing is to combine error resilience with
universal computation. Diagonal gates such as the transversal $T$ gate play an
important role in implementing a universal set of quantum operations. This
paper introduces a framework that describes the process of preparing a code
state, applying a diagonal physical gate, measuring a code syndrome, and
applying a Pauli correction that may depend on the measured syndrome (the
average logical channel induced by an arbitrary diagonal gate). It focuses on
CSS codes, and describes the interaction of code states and physical gates in
terms of generator coefficients determined by the induced logical operator. The
interaction of code states and diagonal gates depends very strongly on the
signs of $Z$-stabilizers in the CSS code, and the proposed generator
coefficient framework explicitly includes this degree of freedom. The paper
derives necessary and sufficient conditions for an arbitrary diagonal gate to
preserve the code space of a stabilizer code, and provides an explicit
expression of the induced logical operator. When the diagonal gate is a
quadratic form diagonal gate (introduced by Rengaswamy et al.), the conditions
can be expressed in terms of divisibility of weights in the two classical codes
that determine the CSS code. These codes find application in magic state
distillation and elsewhere. When all the signs are positive, the paper
characterizes all possible CSS codes, invariant under transversal $Z$-rotation
through $\pi/2^l$, that are constructed from classical Reed-Muller codes by
deriving the necessary and sufficient constraints on $l$. The generator
coefficient framework extends to arbitrary stabilizer codes but there is
nothing to be gained by considering the more general class of non-degenerate
stabilizer codes.
