287 research outputs found
Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes
We introduce the notion of the stopping redundancy hierarchy of a linear
block code as a measure of the trade-off between performance and complexity of
iterative decoding for the binary erasure channel. We derive lower and upper
bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and
Bonferroni-type inequalities, and specialize them for codes with cyclic
parity-check matrices. Based on the observed properties of parity-check
matrices with good stopping redundancy characteristics, we develop a novel
decoding technique, termed automorphism group decoding, that combines iterative
message passing and permutation decoding. We also present bounds on the
smallest number of permutations of an automorphism group decoder needed to
correct any set of erasures up to a prescribed size. Simulation results
demonstrate that for a large number of algebraic codes, the performance of the
new decoding method is close to that of maximum likelihood decoding.Comment: 40 pages, 6 figures, 10 tables, submitted to IEEE Transactions on
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Rate-Compatible Polar Codes for Automorphism Ensemble Decoding
Recently, automorphism ensemble decoding (AED) has drawn research interest as
a more computationally efficient alternative to successive cancellation list
(SCL) decoding of polar codes. Although AED has demonstrated superior
performance for specific code parameters, a flexible code design that can
accommodate varying code rates does not yet exist. This work proposes a
theoretical framework for constructing rate-compatible polar codes with a
prescribed automorphism group, which is a key requirement for AED. We first
prove that a one-bit granular sequence with useful automorphisms cannot exist.
However, by allowing larger steps in the code dimension, flexible code
sequences can be constructed. An explicit synthetic channel ranking based on
the -expansion is then proposed to ensure that all constructed codes
possess the desired symmetries. Simulation results, covering a broad range of
code dimensions and blocklengths, show a performance comparable to that of 5G
polar codes under cyclic redundancy check (CRC)-aided SCL decoding, however,
with lower complexity.Comment: 5 pages, 2 figures, submitted to IEEE for possible publicatio
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