172 research outputs found
Existence and Construction of LCD codes over Finite Fields
We demonstrate the existence of Euclidean and Hermitian LCD codes over finite
fields with various parameters. In addition, we provide a method for
constructing multiple Hermitian LCD (self orthogonal, self dual) codes from a
given Hermitian LCD (self orthogonal, self dual) code, as well as a method for
constructing Euclidean (Hermitian) LCD codes with parameters and
from a given Euclidean (Hermitian) LCD code with parameters
over finite fields. Finally, we provide some findings on -LCD codes
over finite fields
and
The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(pe, l) (including Zpe). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF (2e, l) of length n = 2l for any a β₯ 1 and l β₯ 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem
Intersections of linear codes and related MDS codes with new Galois hulls
Let denote the group of all semilinear
isometries on , where is a prime power. In this
paper, we investigate general properties of linear codes associated with
duals for . We show that
the dimension of the intersection of two linear codes can be determined by
generator matrices of such codes and their duals. We also show that
the dimension of hull of a linear code can be determined by a
generator matrix of it or its dual. We give a characterization on
dual and hull of a matrix-product code. We also investigate
the intersection of a pair of matrix-product codes. We provide a necessary and
sufficient condition under which any codeword of a generalized Reed-Solomon
(GRS) code or an extended GRS code is contained in its dual. As an
application, we construct eleven families of -ary MDS codes with new
-Galois hulls satisfying , which are not covered by the
latest papers by Cao (IEEE Trans. Inf. Theory 67(12), 7964-7984, 2021) and by
Fang et al. (Cryptogr. Commun. 14(1), 145-159, 2022) when
- β¦