Abstract-This paper considers the problem of variable-length coding over a discrete memoryless channel (DMC) with noiseless feedback. The paper provides a stochastic control view of the problem whose solution is analyzed via a newly proposed symmetrized divergence, termed extrinsic Jensen-Shannon (EJS) divergence. It is shown that strictly positive lower bounds on EJS divergence provide non-asymptotic upper bounds on the expected code length. The paper presents strictly positive lower bounds on EJS divergence, and hence non-asymptotic upper bounds on the expected code length, for the following two coding schemes: variable-length posterior matching and MaxEJS coding scheme which is based on a greedy maximization of the EJS divergence. As an asymptotic corollary of the main results, this paper also provides a rate-reliability test. Variable-length coding schemes that satisfy the condition(s) of the test for parameters R and E, are guaranteed to achieve rate R and error exponent E. The results are specialized for posterior matching and MaxEJS to obtain deterministic one-phase coding schemes achieving capacity and optimal error exponent. For the special case of symmetric binary-input channels, simpler deterministic schemes of optimal performance are proposed and analyzed. Index Terms-Discrete memoryless channel, variable-length coding, sequential analysis, feedback gain, Burnashev's reliability function, optimal error exponent.
Naghsvar and Tavidi considered variable-length encoding with feedback and utilized principles from stochastic control and posterior matching to demonstrate non-asymptotic upper bounds on expected code length and deterministic one-phase coding schemes that achieve capacity and attain optimal error exponents REF .
Extrinsic Jensen-Shannon Divergence: Applications to Variable-Length Coding
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Mathematics Computer Science