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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
Skew-cyclic codes
We generalize the notion of cyclic codes by using generator polynomials in
(non commutative) skew polynomial rings. Since skew polynomial rings are left
and right euclidean, the obtained codes share most properties of cyclic codes.
Since there are much more skew-cyclic codes, this new class of codes allows to
systematically search for codes with good properties. We give many examples of
codes which improve the previously best known linear codes
Skew Cyclic codes over \F_q+u\F_q+v\F_q+uv\F_q
In this paper, we study skew cyclic codes over the ring
R=\F_q+u\F_q+v\F_q+uv\F_q, where , and
is an odd prime. We investigate the structural properties of skew cyclic codes
over through a decomposition theorem. Furthermore, we give a formula for
the number of skew cyclic codes of length over $R.
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