  Consider a relay cascade, i.e. a network where a source node, a sink node and
a certain number of intermediate source/relay nodes are arranged on a line and
where adjacent node pairs are connected by error-free (q+1)-ary pipes. Suppose
the source and a subset of the relays wish to communicate independent
information to the sink under the condition that each relay in the cascade is
half-duplex constrained. A coding scheme is developed which transfers
information by an information-dependent allocation of the transmission and
reception slots of the relays. The coding scheme requires synchronization on
the symbol level through a shared clock. The coding strategy achieves capacity
for a single source. Numerical values for the capacity of cascades of various
lengths are provided, and the capacities are significantly higher than the
rates which are achievable with a predetermined time-sharing approach. If the
cascade includes a source and a certain number of relays with their own
information, the strategy achieves the cut-set bound when the rates of the
relay sources fall below certain thresholds. For cascades composed of an
infinite number of half-duplex constrained relays and a single source, we
derive an explicit capacity expression. Remarkably, the capacity in bits/use
for q=1 is equal to the logarithm of the golden ratio, and the capacity for q=2
is 1 bit/use.
