26,584 research outputs found
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
Properties and classifications of certain LCD codes.
A linear code is called a linear complementary dual code (LCD code) if holds. LCD codes have many applications in cryptography, communication systems, data storage, and quantum coding theory. In this dissertation we show that a necessary and sufficient condition for a cyclic code over of odd length to be an LCD code is that where is a self-reciprocal polynomial in which is also in our paper \cite{GK1}. We then extend this result and provide a necessary and sufficient condition for a cyclic code of length over a finite chain ring R=\big(R,\m=(\gamma),\kappa=R/\m \big) with to be an LCD code. In \cite{DKOSS} a linear programming bound for LCD codes and the definition for for binary LCD -codes are provided. Thus, in a different direction, we find the formula for which appears in \cite{GK2}. In 2020, Pang et al. defined binary codes with biggest minimal distance, which meets the Griesmer bound \cite{Pang}. We give a correction to and provide a different proof for \cite[Theorem 4.2]{Pang}, provide a different proof for \cite[Theorem 4.3]{Pang}, examine properties of LCD ternary codes, and extend some results found in \cite{Harada} for any which is a power of an odd prime
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
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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