  Landmark codes underpin reliable physical layer communication, e.g.,
Reed-Muller, BCH, Convolution, Turbo, LDPC and Polar codes: each is a linear
code and represents a mathematical breakthrough. The impact on humanity is
huge: each of these codes has been used in global wireless communication
standards (satellite, WiFi, cellular). Reliability of communication over the
classical additive white Gaussian noise (AWGN) channel enables benchmarking and
ranking of the different codes. In this paper, we construct KO codes, a
computationaly efficient family of deep-learning driven (encoder, decoder)
pairs that outperform the state-of-the-art reliability performance on the
standardized AWGN channel. KO codes beat state-of-the-art Reed-Muller and Polar
codes, under the low-complexity successive cancellation decoding, in the
challenging short-to-medium block length regime on the AWGN channel. We show
that the gains of KO codes are primarily due to the nonlinear mapping of
information bits directly to transmit real symbols (bypassing modulation) and
yet possess an efficient, high performance decoder. The key technical
innovation that renders this possible is design of a novel family of neural
architectures inspired by the computation tree of the {\bf K}ronecker {\bf
O}peration (KO) central to Reed-Muller and Polar codes. These architectures
pave way for the discovery of a much richer class of hitherto unexplored
nonlinear algebraic structures. The code is available at
\href{https://github.com/deepcomm/KOcodes}{https://github.com/deepcomm/KOcodes}
