5 research outputs found
Fast syndrome-based Chase decoding of binary BCH codes through Wu list decoding
We present a new fast Chase decoding algorithm for binary BCH codes. The new
algorithm reduces the complexity in comparison to a recent fast Chase decoding
algorithm for Reed--Solomon (RS) codes by the authors (IEEE Trans. IT, 2022),
by requiring only a single Koetter iteration per edge of the decoding tree. In
comparison to the fast Chase algorithms presented by Kamiya (IEEE Trans. IT,
2001) and Wu (IEEE Trans. IT, 2012) for binary BCH codes, the polynomials
updated throughout the algorithm of the current paper typically have a much
lower degree.
To achieve the complexity reduction, we build on a new isomorphism between
two solution modules in the binary case, and on a degenerate case of the
soft-decision (SD) version of the Wu list decoding algorithm. Roughly speaking,
we prove that when the maximum list size is in Wu list decoding of binary
BCH codes, assigning a multiplicity of to a coordinate has the same effect
as flipping this coordinate in a Chase-decoding trial.
The solution-module isomorphism also provides a systematic way to benefit
from the binary alphabet for reducing the complexity in bounded-distance
hard-decision (HD) decoding. Along the way, we briefly develop the
Groebner-bases formulation of the Wu list decoding algorithm for binary BCH
codes, which is missing in the literature
Efficient Algorithms for Constructing Minimum-Weight Codewords in Some Extended Binary BCH Codes
We present algorithms for specifying the support of minimum-weight
words of extended binary BCH codes of length and designed distance
for some values of , where may
grow to infinity. The support is specified as the sum of two sets: a set of
elements, and a subspace of dimension , specified by
a basis.
In some detail, for designed distance , we have a deterministic
algorithm for even , and a probabilistic algorithm with success
probability for odd . For designed distance ,
we have a probabilistic algorithm with success probability for even . Finally, for designed distance , we have a deterministic algorithm for divisible by . We also
present a construction via Gold functions when .
Our construction builds on results of Kasami and Lin (IEEE T-IT, 1972), who
proved that for extended binary BCH codes of designed distance , the
minimum distance equals the designed distance. Their proof makes use of a
non-constructive result of Berlekamp (Inform. Contrl., 1970), and a
constructive ``down-conversion theorem'' that converts some words in BCH codes
to lower-weight words in BCH codes of lower designed distance. Our main
contribution is in replacing the non-constructive argument of Berlekamp by a
low-complexity algorithm.
In one aspect, we extends the results of Grigorescu and Kaufman (IEEE T-IT,
2012), who presented explicit minimum-weight words for designed distance
(and hence also for designed distance , by a well-known
``up-conversion theorem''), as we cover more cases of the minimum distance.
However, the minimum-weight words we construct are not affine generators for
designed distance