  For improving short-length codes, we demonstrate that classic decoders can
also be used with real-valued, neural encoders, i.e., deep-learning based
codeword sequence generators. Here, the classical decoder can be a valuable
tool to gain insights into these neural codes and shed light on weaknesses.
Specifically, the turbo-autoencoder is a recently developed channel coding
scheme where both encoder and decoder are replaced by neural networks. We first
show that the limited receptive field of convolutional neural network
(CNN)-based codes enables the application of the BCJR algorithm to optimally
decode them with feasible computational complexity. These maximum a posteriori
(MAP) component decoders then are used to form classical (iterative) turbo
decoders for parallel or serially concatenated CNN encoders, offering a
close-to-maximum likelihood (ML) decoding of the learned codes. To the best of
our knowledge, this is the first time that a classical decoding algorithm is
applied to a non-trivial, real-valued neural code. Furthermore, as the BCJR
algorithm is fully differentiable, it is possible to train, or fine-tune, the
neural encoder in an end-to-end fashion.
