  To well understand the behavior of quantum error correction codes (QECC) in
noise processes, we need to obtain explicit coding maps for QECC. Due to
extraordinary amount of computational labor that they entails, explicit coding
maps are a little known. Indeed this is even true for one of the most commonly
considered quantum codes-the five-qubit code, also known as the smallest
perfect code that permits corrections of generic single-qubit errors. With
direct but complicated computation, we obtain explicit process matrix of the
coding maps with a unital error channel for the five-qubit code. The process
matrix allows us to conduct exact analysis on the performance of the quantum
code. We prove that the code can correct a generic error in the sense that
under repeated concatenation of the coding map with itself, the code does not
make any assumption about the error model other than it being weak and thus can
remove the error(it can transform/take the error channel to the identity
channel if the error is sufficiently small.). We focus on the examination of
some coherent error models (non diagonal channels) studied in recent
literatures. We numerically derive a lower bound on threshold of the
convergence for the code. Furthermore, we analytically show how the code
affects the average gate infidelity and diamond distance of the error channels.
Explicit formulas of the two measurements for both pre-error channel and
post-error channel are derived, and we then analyze the logical error rates of
the aforesaid quantum code. Our findings tighten the upper bounds on diamond
distance of the noise channel after error corrections obtained in literature.
