  We introduce a discrete network corresponding to any Gaussian wireless
network that is obtained by simply quantizing the received signals and
restricting the transmitted signals to a finite precision. Since signals in the
discrete network are obtained from those of a Gaussian network, the Gaussian
network can be operated on the quantization-based digital interface defined by
the discrete network. We prove that this digital interface is near-optimal for
Gaussian relay networks and the capacities of the Gaussian and the discrete
networks are within a bounded gap of O(M^2) bits, where M is the number of
nodes.
  We prove that any near-optimal coding strategy for the discrete network can
be naturally transformed into a near-optimal coding strategy for the Gaussian
network merely by quantization. We exploit this by designing a linear coding
strategy for the case of layered discrete relay networks. The linear coding
strategy is near-optimal for Gaussian and discrete networks and achieves rates
within O(M^2) bits of the capacity, independent of channel gains or SNR. The
linear code is robust and the relays need not know the channel gains. The
transmit and receive signals at all relays are simply quantized to binary
tuples of the same length $n$ . The linear network code requires all the relay
nodes to collect the received binary tuples into a long binary vector and apply
a linear transformation on the long vector. The resulting binary vector is
split into smaller binary tuples for transmission by the relays. The
quantization requirements of the linear network code are completely defined by
the parameter $n$, which also determines the resolution of the
analog-to-digital and digital-to-analog convertors for operating the network
within a bounded gap of the network's capacity. The linear network code
explicitly connects network coding for wireline networks with codes for
Gaussian networks.
