  We introduce a new class of circuits for constructing efficiently decodable
error-correction codes, based on a recently discovered contractible tensor
network. We perform an in-depth study of a particular example that can be
thought of as an extension to Arikan's polar code. Notably, our numerical
simulation show that this code polarizes the logical channels more strongly
while retaining the log-linear decoding complexity using the successive
cancellation decoder. These codes also display improved error-correcting
capability with only a minor impact on decoding complexity. Efficient decoding
is realized using powerful graphical calculus tools developed in the field of
quantum many-body physics. In a companion paper, we generalize our construction
to the quantum setting and describe more in-depth the relation between
classical and quantum error correction and the graphical calculus of tensor
networks.
