  Existing fixed-length feedback communication schemes are either specialized
to particular channels (Schalkwijk--Kailath, Horstein), or apply to general
channels but either have high coding complexity (block feedback schemes) or are
difficult to analyze (posterior matching). This paper introduces a new
fixed-length feedback coding scheme which achieves the capacity for all
discrete memoryless channels, has an error exponent that approaches the sphere
packing bound as the rate approaches the capacity, and has $O(n\log n)$ coding
complexity. These benefits are achieved by judiciously combining features from
previous schemes with new randomization technique and encoding/decoding rule.
These new features make the analysis of the error probability for the new
scheme easier than for posterior matching.
