134 research outputs found
Rank equivalent and rank degenerate skew cyclic codes
Two skew cyclic codes can be equivalent for the Hamming metric only if they
have the same length, and only the zero code is degenerate. The situation is
completely different for the rank metric, where lengths of codes correspond to
the number of outgoing links from the source when applying the code on a
network. We study rank equivalences between skew cyclic codes of different
lengths and, with the aim of finding the skew cyclic code of smallest length
that is rank equivalent to a given one, we define different types of length for
a given skew cyclic code, relate them and compute them in most cases. We give
different characterizations of rank degenerate skew cyclic codes using
conventional polynomials and linearized polynomials. Some known results on the
rank weight hierarchy of cyclic codes for some lengths are obtained as
particular cases and extended to all lengths and to all skew cyclic codes.
Finally, we prove that the smallest length of a linear code that is rank
equivalent to a given skew cyclic code can be attained by a pseudo-skew cyclic
code. Throughout the paper, we find new relations between linear skew cyclic
codes and their Galois closures
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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