  A status updating system is considered in which a variable length code is
used to transmit messages to a receiver over a noisy channel. The goal is to
optimize the codewords lengths such that successfully-decoded messages are
timely. That is, such that the age-of-information (AoI) at the receiver is
minimized. A hybrid ARQ (HARQ) scheme is employed, in which variable-length
incremental redundancy (IR) bits are added to the originally-transmitted
codeword until decoding is successful. With each decoding attempt, a non-zero
processing delay is incurred. The optimal codewords lengths are analytically
derived utilizing a sequential differential optimization (SDO) framework. The
framework is general in that it only requires knowledge of an analytical
expression of the positive feedback (ACK) probability as a function of the
codeword length.
