  The design of optimal linear block codes capable of being efficiently decoded
is of major concern, especially for short block lengths. As near
capacity-approaching codes, Low-Density Parity-Check (LDPC) codes possess
several advantages over other families of codes, the most notable being its
efficient decoding via Belief Propagation. While many LDPC code design methods
exist, the development of efficient sparse codes that meet the constraints of
modern short code lengths and accommodate new channel models remains a
challenge. In this work, we propose for the first time a data-driven approach
for the design of sparse codes. We develop locally optimal codes with respect
to Belief Propagation decoding via the learning on the Factor graph (also
called the Tanner graph) under channel noise simulations. This is performed via
a novel tensor representation of the Belief Propagation algorithm, optimized
over finite fields via backpropagation coupled with an efficient line-search
method. The proposed approach is shown to outperform the decoding performance
of existing popular codes by orders of magnitude and demonstrates the power of
data-driven approaches for code design.
