  Dynamical decoupling of spin qubits in silicon can enhance fidelity and be
used to extract the frequency spectra of noise processes. We demonstrate a
full-permutation dynamical decoupling technique that cyclically exchanges the
spins in a triple-dot qubit. This sequence not only suppresses both low
frequency charge-noise- and magnetic-noise-induced errors; it also refocuses
leakage errors to first order, which is particularly interesting for encoded
exchange-only qubits. For a specific construction, which we call NZ1y, the
qubit is isolated from error sources to such a degree that we measure a
remarkable exchange pulse error of $5\times10^{-5}$. This sequence maintains a
quantum state for roughly 18,000 exchange pulses, extending the qubit coherence
from $T_2^*=2~\mu$s to $T_2 = 720~\mu$s. We experimentally validate an error
model that includes $1/f$ charge noise and $1/f$ magnetic noise in two ways: by
direct exchange-qubit simulation, and by integration of the assumed noise
spectra with derived filter functions, both of which reproduce the measured
error and leakage with respect to changing the repetition rate.
