  In order to perform universal fault-tolerant quantum computation, one needs
to implement a logical non-Clifford gate. Consequently, it is important to
understand codes that implement such gates transversally. In this paper, we
adopt an algebraic approach to characterize all stabilizer codes for which
transversal $T$ and $T^{-1}$ gates preserve the codespace. Our Heisenberg
perspective reduces this to a finite geometry problem that translates to the
design of certain classical codes. We prove three corollaries: (a) For any
non-degenerate $[[ n,k,d ]]$ stabilizer code supporting a physical transversal
$T$, there exists an $[[ n,k,d ]]$ CSS code with the same property; (b)
Triorthogonal codes are the most general CSS codes that realize logical
transversal $T$ via physical transversal $T$; (c) Triorthogonality is necessary
for physical transversal $T$ on a CSS code to realize the logical identity. The
main tool we use is a recent efficient characterization of certain diagonal
gates in the Clifford hierarchy (arXiv:1902.04022). We refer to these gates as
Quadratic Form Diagonal (QFD) gates. Our framework generalizes all existing
code constructions that realize logical gates via transversal $T$. We provide
several examples and briefly discuss connections to decreasing monomial codes,
pin codes, generalized triorthogonality and quasitransversality. We partially
extend these results towards characterizing all stabilizer codes that support
transversal $\pi/2^{\ell}$ $Z$-rotations. In particular, using Ax's theorem on
residue weights of polynomials, we provide an alternate characterization of
logical gates induced by transversal $\pi/2^{\ell}$ $Z$-rotations on a family
of quantum Reed-Muller codes. We also briefly discuss a general approach to
analyze QFD gates that might lead to a characterization of all stabilizer codes
that support any given physical transversal $1$- or $2$-local diagonal gate.
