78 research outputs found
Skew-constacyclic codes over
In this paper, the investigation on the algebraic structure of the ring
and the description of its
automorphism group, enable to study the algebraic structure of codes and their
dual over this ring. We explore the algebraic structure of skew-constacyclic
codes, by using a linear Gray map and we determine their generator polynomials.
Necessary and sufficient conditions for the existence of self-dual skew cyclic
and self-dual skew negacyclic codes over
are given
A note on the duals of skew constacyclic codes
Let be a finite field with elements and denote by an automorphism of . In this
paper, we deal with skew constacyclic codes, that is, linear codes of
which are invariant under the action of a semi-linear map
, defined by
for some
and . In particular, we study
some algebraic and geometric properties of their dual codes and we give some
consequences and research results on -generator skew quasi-twisted codes and
on MDS skew constacyclic codes.Comment: 31 pages, 3 tables; this is a revised version that includes
improvements to the presentation of the main results, a new subsection and an
appendix which is an extension of Section 2 of the previous versio
On the codes over the Z_3+vZ_3+v^2Z_3
In this paper, we study the structure of cyclic, quasi-cyclic, constacyclic
codes and their skew codes over the finite ring R=Z_3+vZ_3+v^2Z_3, v^3=v. The
Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasi-cyclic and skew
constacyclic codes over R are obtained. A necessary and sufficient condition
for cyclic (negacyclic) codes over R that contains its dual has been given. The
parameters of quantum error correcting codes are obtained from both cyclic and
negacyclic codes over R. It is given some examples. Firstly, quasi-constacyclic
and skew quasi-constacyclic codes are introduced. By giving two product, it is
investigated their duality. A sufficient condition for 1-generator skew
quasi-constacyclic codes to be free is determined
Skew Constacyclic Codes over Finite Chain Rings
Skew polynomial rings over finite fields ([7] and [10]) and over Galois rings
([8]) have been used to study codes. In this paper, we extend this concept to
finite chain rings. Properties of skew constacyclic codes generated by monic
right divisors of , where is a unit element, are
exhibited. When , the generators of Euclidean and Hermitian dual
codes of such codes are determined together with necessary and sufficient
conditions for them to be Euclidean and Hermitian self-dual. Of more interest
are codes over the ring . The structure of
all skew constacyclic codes is completely determined. This allows us to express
generators of Euclidean and Hermitian dual codes of skew cyclic and skew
negacyclic codes in terms of the generators of the original codes. An
illustration of all skew cyclic codes of length~2 over
and their Euclidean and Hermitian dual codes
is also provided.Comment: 24 Pages, Submitted to Advances in Mathematics of Communication
Skew constacyclic codes over Fq+uFq+vFq
In this paper skew constacyclic codes over finite non-chain ring R =
F_q+uF_q+vF_q, where q= p^m, p is an odd prime and u^{2}=u, v^{2}=v, uv = vu =
0 are studied. We show that Gray image of a skew alpha-constacyclic cyclic code
of length n over R is a skew alpha-quasi-cyclic code of length over F_{q}
of index 3. It is also shown that skew alpha-constacyclic codes are either
equivalent to alpha-constacyclic codes or alpha-quasi-twisted codes over R.
Further, the structural properties of skew constacyclic over R are obtained by
decomposition method.Comment: 10 pages paper communicated to the Journal of Algebra and its
Application
Skew constacyclic codes over a non-chain ring
Let and be two polynomials of degree and
respectively, not both linear, which split into distinct linear factors over
. Let be a finite commutative non-chain ring. In this
paper, we study -skew cyclic and -skew constacyclic codes over
the ring where and are two automorphisms
defined on .Comment: 15 page
-Linear Skew Constacyclic Codes
In this paper, we study skew constacyclic codes over the ring
where , for a
prime and . We give the definition of these codes as subsets of
the ring . Some structural properties of the
skew polynomial ring are discussed, where is an
automorphism of . We describe the generator polynomials of skew constacyclic
codes over and . Using Gray images of skew constacyclic
codes over we obtained some new linear codes over
. Further, we have generalized these codes to double skew
constacyclic codes over
On Principal -Codes over Rings
Let be a ring with identity, a ring endomorphism of that
maps the identity to itself, a -derivation of , and
consider the skew-polynomial ring . When is a finite
field, a Galois ring, or a general ring, some fairly recent literature used
to construct new interesting codes (e.g. skew-cyclic and
skew-constacyclic codes) that generalize their classical counterparts over
finite fields (e.g. cyclic and constacyclic linear codes). This paper presents
results concerning {\it principal} -codes over a ring ,
where is monic. We provide recursive formulas that
compute the entries of both a generating matrix and a control matrix of such a
code . When is a finite commutative ring with identity and
is a ring automorphism of , we also give recursive formulas for the
entries of a parity-check matrix of . Also in this case, with
, we give a generating matrix of the dual ,
present a characterization of principal -codes whose duals are also
principal -codes, and deduce a characterization of self-dual principal
-codes. Some corollaries concerning principal -constacyclic
codes are also given, and some highlighting examples are provided.Comment: 17 page
New Quantum MDS codes constructed from Constacyclic codes
Quantum maximum-distance-separable (MDS) codes are an important class of
quantum codes. In this paper, using constacyclic codes and Hermitain
construction, we construct some new quantum MDS codes of the form ,
. Most of these quantum MDS codes are new in the sense
that their parameters are not covered be the codes available in the literature.Comment: arXiv admin note: text overlap with arXiv:1803.0416
Entanglement-assisted quantum MDS codes from constacyclic codes with large minimum distance
The entanglement-assisted (EA) formalism allows arbitrary classical linear
codes to transform into entanglement-assisted quantum error correcting codes
(EAQECCs) by using pre-shared entanglement between the sender and the receiver.
In this work, we propose a decomposition of the defining set of constacyclic
codes. Using this method, we construct four classes of -ary
entanglement-assisted quantum MDS (EAQMDS) codes based on classical
constacyclic MDS codes by exploiting less pre-shared maximally entangled
states. We show that a class of -ary EAQMDS have minimum distance upper
limit greater than . Some of them have much larger minimum distance than
the known quantum MDS (QMDS) codes of the same length. Most of these -ary
EAQMDS codes are new in the sense that their parameters are not covered by the
codes available in the literature
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