8,693 research outputs found
Concatenated structure and construction of certain code families
In this thesis, we consider concatenated codes and their generalizations as the main tool for two different purposes. Our first aim is to extend the concatenated structure of quasi-cyclic codes to its two generalizations: generalized quasi-cyclic codes and quasi-abelian codes. Concatenated structure have consequences such as a general minimum distance bound. Hence, we obtain minimum distance bounds, which are analogous to Jensen's bound for quasi-cyclic codes, for generalized quasicyclic and quasi-abelian codes. We also prove that linear complementary dual quasi-abelian codes are asymptotically good, using the concatenated structure. Moreover, for generalized quasi-cyclic and quasi-abelian codes, we prove, as in the quasi-cyclic codes, that their concatenated decomposition and the Chinese Remainder decomposition are equivalent. The second purpose of the thesis is to construct a linear complementary pair of codes using concatenations. This class of codes have been of interest recently due to their applications in cryptography. This extends the recent result of Carlet et al. on the concatenated construction of linear complementary dual codes
Some quaternary additive codes outperform linear counterparts
The additive codes may have better parameters than linear codes. However, it
is still a challenging problem to efficiently construct additive codes that
outperform linear codes, especially those with greater distances than linear
codes of the same lengths and dimensions. This paper focuses on constructing
additive codes that outperform linear codes based on quasi-cyclic codes and
combinatorial methods. Firstly, we propose a lower bound on the symplectic
distance of 1-generator quasi-cyclic codes of index even. Secondly, we get many
binary quasi-cyclic codes with large symplectic distances utilizing
computer-supported combination and search methods, all of which correspond to
good quaternary additive codes. Notably, some additive codes have greater
distances than best-known quaternary linear codes in Grassl's code table
(bounds on the minimum distance of quaternary linear codes
http://www.codetables.de) for the same lengths and dimensions. Moreover,
employing a combinatorial approach, we partially determine the parameters of
optimal quaternary additive 3.5-dimensional codes with lengths from to
. Finally, as an extension, we also construct some good additive
complementary dual codes with larger distances than the best-known quaternary
linear complementary dual codes in the literature
On Euclidean, Hermitian and symplectic quasi-cyclic complementary dual codes
Linear complementary dual codes (LCD) intersect trivially with their dual. In
this paper, we develop a new characterization for LCD codes, which allows us to
judge the complementary duality of linear codes from the codeword level.
Further, we determine the sufficient and necessary conditions for one-generator
quasi-cyclic codes to be LCD codes involving Euclidean, Hermitian, and
symplectic inner products. Finally, we constructed many Euclidean, Hermitian
and symmetric LCD codes with excellent parameters, some improving the results
in the literature. Remarkably, we construct a symplectic LCD code
with symplectic distance , which corresponds to an trace Hermitian additive
complementary dual code that outperforms the optimal quaternary
Hermitian LCD code
Quaternary Conjucyclic Codes with an Application to EAQEC Codes
Conjucyclic codes are part of a family of codes that includes cyclic,
constacyclic, and quasi-cyclic codes, among others. Despite their importance in
quantum error correction, they have not received much attention in the
literature. This paper focuses on additive conjucyclic (ACC) codes over
and investigates their properties. Specifically, we derive the
duals of ACC codes using a trace inner product and obtain the trace hull and
its dimension. Also, establish a necessary and sufficient condition for an
additive code to have a complementary dual (ACD). Additionally, we identify a
necessary condition for an additive conjucyclic complementary pair of codes
over . Furthermore, we show that the trace code of an ACC code is
cyclic and provide a condition for the trace code of an ACC code to be LCD. To
demonstrate the practical application of our findings, we construct some good
entanglement-assisted quantum error-correcting (EAQEC) codes using the trace
code of ACC codes
Quasi-Cyclic Complementary Dual Code
LCD codes are linear codes that intersect with their dual trivially. Quasi
cyclic codes that are LCD are characterized and studied by using their
concatenated structure. Some asymptotic results are derived. Hermitian LCD
codes are introduced to that end and their cyclic subclass is characterized.
Constructions of QCCD codes from codes over larger alphabets are given
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