227 research outputs found
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
On MDS Negacyclic LCD Codes
Linear codes with complementary duals (LCD) have a great deal of significance
amongst linear codes. Maximum distance separable (MDS) codes are also an
important class of linear codes since they achieve the greatest error
correcting and detecting capabilities for fixed length and dimension. The
construction of linear codes that are both LCD and MDS is a hard task in coding
theory. In this paper, we study the constructions of LCD codes that are MDS
from negacyclic codes over finite fields of odd prime power elements. We
construct four families of MDS negacyclic LCD codes of length
, and a family of negacyclic LCD codes
of length . Furthermore, we obtain five families of -ary
Hermitian MDS negacyclic LCD codes of length and four
families of Hermitian negacyclic LCD codes of length For both
Euclidean and Hermitian cases the dimensions of these codes are determined and
for some classes the minimum distances are settled. For the other cases, by
studying and -cyclotomic classes we give lower bounds on the minimum
distance
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