8,408 research outputs found
Symplectic self-orthogonal quasi-cyclic codes
In this paper, we obtain sufficient and necessary conditions for quasi-cyclic
codes with index even to be symplectic self-orthogonal. Then, we propose a
method for constructing symplectic self-orthogonal quasi-cyclic codes, which
allows arbitrary polynomials that coprime to construct symplectic
self-orthogonal codes. Moreover, by decomposing the space of quasi-cyclic
codes, we provide lower and upper bounds on the minimum symplectic distances of
a class of 1-generator quasi-cyclic codes and their symplectic dual codes.
Finally, we construct many binary symplectic self-orthogonal codes with
excellent parameters, corresponding to 117 record-breaking quantum codes,
improving Grassl's table (Bounds on the Minimum Distance of Quantum Codes.
http://www.codetables.de)
On the Structure of the Linear Codes with a Given Automorphism
The purpose of this paper is to present the structure of the linear codes
over a finite field with q elements that have a permutation automorphism of
order m. These codes can be considered as generalized quasi-cyclic codes.
Quasi-cyclic codes and almost quasi-cyclic codes are discussed in detail,
presenting necessary and sufficient conditions for which linear codes with such
an automorphism are self-orthogonal, self-dual, or linear complementary dual
Quasi-cyclic Hermitian construction of binary quantum codes
In this paper, we propose a sufficient condition for a family of 2-generator
self-orthogonal quasi-cyclic codes with respect to Hermitian inner product.
Supported in the Hermitian construction, we show algebraic constructions of
good quantum codes. 30 new binary quantum codes with good parameters improving
the best-known lower bounds on minimum distance in Grassl's code tables
\cite{Grassl:codetables} are constructed
Generator polynomial matrices of the Galois hulls of multi-twisted codes
In this study, we consider the Euclidean and Galois hulls of multi-twisted
(MT) codes over a finite field of characteristic . Let
be a generator polynomial matrix (GPM) of a MT code .
For any , the -Galois hull of , denoted by
, is the intersection of with
its -Galois dual. The main result in this paper is that a GPM for
has been obtained from . We
start by associating a linear code with .
We show that is quasi-cyclic. In addition, we prove
that the dimension of is the difference
between the dimension of and that of .
Thus the determinantal divisors are used to derive a formula for the dimension
of . Finally, we deduce a GPM formula for
. In particular, we handle the cases of
-Galois self-orthogonal and linear complementary dual MT codes; we
establish equivalent conditions that characterize these cases. Equivalent
results can be deduced immediately for the classes of cyclic, constacyclic,
quasi-cyclic, generalized quasi-cyclic, and quasi-twisted codes, because they
are all special cases of MT codes. Some numerical examples, containing optimal
and maximum distance separable codes, are used to illustrate the theoretical
results
A Class of Quantum LDPC Codes Constructed From Finite Geometries
Low-density parity check (LDPC) codes are a significant class of classical
codes with many applications. Several good LDPC codes have been constructed
using random, algebraic, and finite geometries approaches, with containing
cycles of length at least six in their Tanner graphs. However, it is impossible
to design a self-orthogonal parity check matrix of an LDPC code without
introducing cycles of length four.
In this paper, a new class of quantum LDPC codes based on lines and points of
finite geometries is constructed. The parity check matrices of these codes are
adapted to be self-orthogonal with containing only one cycle of length four.
Also, the column and row weights, and bounds on the minimum distance of these
codes are given. As a consequence, the encoding and decoding algorithms of
these codes as well as their performance over various quantum depolarizing
channels will be investigated.Comment: 5pages, 2 figure
Euclidean and Hermitian LCD MDS codes
Linear codes with complementary duals (abbreviated LCD) are linear codes
whose intersection with their dual is trivial. When they are binary, they play
an important role in armoring implementations against side-channel attacks and
fault injection attacks. Non-binary LCD codes in characteristic 2 can be
transformed into binary LCD codes by expansion. On the other hand, being
optimal codes, maximum distance separable codes (abbreviated MDS) have been of
much interest from many researchers due to their theoretical significant and
practical implications. However, little work has been done on LCD MDS codes. In
particular, determining the existence of -ary LCD MDS codes for
various lengths and dimensions is a basic and interesting problem. In
this paper, we firstly study the problem of the existence of -ary
LCD MDS codes and completely solve it for the Euclidean case. More
specifically, we show that for there exists a -ary Euclidean
LCD MDS code, where , or, , and . Secondly, we investigate several constructions of new Euclidean
and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and
Hermitian LCD MDS codes use some linear codes with small dimension or
codimension, self-orthogonal codes and generalized Reed-Solomon codes
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