145 research outputs found
Research on Hermitian self-dual codes, GRS codes and EGRS codes
MDS self-dual codes have nice algebraic structures, theoretical significance
and practical implications. In this paper, we present three classes of
-ary Hermitian self-dual (extended) generalized Reed-Solomon codes with
different code locators. Combining the results in Ball et al. (Designs, Codes
and Cryptography, 89: 811-821, 2021), we show that if the code locators do not
contain zero, -ary Hermitian self-dual (extended) GRS codes of length
does not exist. Under certain conditions, we prove Conjecture
3.7 and Conjecture 3.13 proposed by Guo and Li et al. (IEEE Communications
Letters, 25(4): 1062-1065, 2021).Comment: 18 page
On Hull-Variation Problem of Equivalent Linear Codes
The intersection () of a linear code and its Euclidean dual (Hermitian dual ) is called the Euclidean
(Hermitian) hull of this code. The construction of an entanglement-assisted
quantum code from a linear code over or depends
essentially on the Euclidean hull or the Hermitian hull of this code. Therefore
it is natural to consider the hull-variation problem when a linear code is transformed to an equivalent code . In this paper
we introduce the maximal hull dimension as an invariant of a linear code with
respect to the equivalent transformations. Then some basic properties of the
maximal hull dimension are studied. A general method to construct
hull-decreasing or hull-increasing equivalent linear codes is proposed. We
prove that for a nonnegative integer satisfying , a
linear self-dual code is equivalent to a linear -dimension hull
code. On the opposite direction we prove that a linear LCD code over satisfying and is equivalent to a linear
one-dimension hull code under a weak condition. Several new families of
negacyclic LCD codes and BCH LCD codes over are also constructed.
Our method can be applied to the generalized Reed-Solomon codes and the
generalized twisted Reed-Solomon codes to construct arbitrary dimension hull
MDS codes. Some new EAQEC codes including MDS and almost MDS
entanglement-assisted quantum codes are constructed. Many EAQEC codes over
small fields are constructed from optimal Hermitian self-dual codes.Comment: 33 pages, minor error correcte
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
Quantum generalized Reed-Solomon codes: Unified framework for quantum MDS codes
We construct a new family of quantum MDS codes from classical generalized
Reed-Solomon codes and derive the necessary and sufficient condition under
which these quantum codes exist. We also give code bounds and show how to
construct them analytically. We find that existing quantum MDS codes can be
unified under these codes in the sense that when a quantum MDS code exists,
then a quantum code of this type with the same parameters also exists. Thus as
far as is known at present, they are the most important family of quantum MDS
codes.Comment: 9 pages, no figure
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