131 research outputs found
Generator polynomial matrices of the Galois hulls of multi-twisted codes
In this study, we consider the Euclidean and Galois hulls of multi-twisted
(MT) codes over a finite field of characteristic . Let
be a generator polynomial matrix (GPM) of a MT code .
For any , the -Galois hull of , denoted by
, is the intersection of with
its -Galois dual. The main result in this paper is that a GPM for
has been obtained from . We
start by associating a linear code with .
We show that is quasi-cyclic. In addition, we prove
that the dimension of is the difference
between the dimension of and that of .
Thus the determinantal divisors are used to derive a formula for the dimension
of . Finally, we deduce a GPM formula for
. In particular, we handle the cases of
-Galois self-orthogonal and linear complementary dual MT codes; we
establish equivalent conditions that characterize these cases. Equivalent
results can be deduced immediately for the classes of cyclic, constacyclic,
quasi-cyclic, generalized quasi-cyclic, and quasi-twisted codes, because they
are all special cases of MT codes. Some numerical examples, containing optimal
and maximum distance separable codes, are used to illustrate the theoretical
results
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
On MDS Codes With Galois Hulls of Arbitrary Dimensions
The Galois hulls of linear codes are a generalization of the Euclidean and
Hermitian hulls of linear codes. In this paper, we study the Galois hulls of
(extended) GRS codes and present several new constructions of MDS codes with
Galois hulls of arbitrary dimensions via (extended) GRS codes. Two general
methods of constructing MDS codes with Galois hulls of arbitrary dimensions by
Hermitian or general Galois self-orthogonal (extended) GRS codes are given.
Using these methods, some MDS codes with larger dimensions and Galois hulls of
arbitrary dimensions can be obtained and relatively strict conditions can also
lead to many new classes of MDS codes with Galois hulls of arbitrary
dimensions.Comment: 21 pages,5 table
On Galois self-orthogonal algebraic geometry codes
Galois self-orthogonal (SO) codes are generalizations of Euclidean and
Hermitian SO codes. Algebraic geometry (AG) codes are the first known class of
linear codes exceeding the Gilbert-Varshamov bound. Both of them have attracted
much attention for their rich algebraic structures and wide applications in
these years. In this paper, we consider them together and study Galois SO AG
codes. A criterion for an AG code being Galois SO is presented. Based on this
criterion, we construct several new classes of maximum distance separable (MDS)
Galois SO AG codes from projective lines and several new classes of Galois SO
AG codes from projective elliptic curves, hyper-elliptic curves and hermitian
curves. In addition, we give an embedding method that allows us to obtain more
MDS Galois SO codes from known MDS Galois SO AG codes.Comment: 18paper
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
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