21,371 research outputs found
Infinite families of 2-designs and 3-designs from linear codes
The interplay between coding theory and -designs started many years ago.
While every -design yields a linear code over every finite field, the
largest for which an infinite family of -designs is derived directly
from a linear or nonlinear code is . Sporadic -designs and -designs
were derived from some linear codes of certain parameters. The major objective
of this paper is to construct many infinite families of -designs and
-designs from linear codes. The parameters of some known -designs are
also derived. In addition, many conjectured infinite families of -designs
are also presented
Infinite families of -designs from a type of five-weight codes
It has been known for a long time that -designs can be employed to
construct both linear and nonlinear codes and that the codewords of a fixed
weight in a code may hold a -design. While a lot of progress in the
direction of constructing codes from -designs has been made, only a small
amount of work on the construction of -designs from codes has been done. The
objective of this paper is to construct infinite families of -designs and
-designs from a type of binary linear codes with five-weights. The total
number of -designs and -designs obtained in this paper are exponential in
any odd and the block size of the designs varies in a huge range.Comment: arXiv admin note: substantial text overlap with arXiv:1607.0481
Linear codes of 2-designs associated with subcodes of the ternary generalized Reed-Muller codes
In this paper, the 3-rank of the incidence matrices of 2-designs supported by
the minimum weight codewords in a family of ternary linear codes considered in
[C. Ding, C. Li, Infinite families of 2-designs and 3-designs from linear
codes, Discrete Mathematics 340(10) (2017) 2415--2431] are computed. A lower
bound on the minimum distance of the ternary codes spanned by the incidence
matrices of these designs is derived, and it is proved that the codes are
subcodes of the 4th order generalized Reed-Muller codes
Infinite families of linear codes supporting more -designs
Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of narrow-sense
BCH codes and their dual codes with and
established that the codewords of the minimum (or the second minimum) weight in
these codes support infinite families of 4-designs or 3-designs. Motivated by
this, we further investigate the codewords of the next adjacent weight in such
codes and discover more infinite classes of -designs with . In
particular, we prove that the codewords of weight in
support -designs when is odd and
-designs when is even, which provide infinite classes of
simple -designs with new parameters. Another significant class of
-designs we produce in this paper has supplementary designs with parameters
4-; these designs have the smallest index among all the
known simple 4- designs derived from codes for prime powers
; and they are further proved to be isomorphic to the 4-designs admitting
the projective general linear group PGL as automorphism group
constructed by Alltop in 1969.Comment: 26 pages, 4 table
Infinite families of -designs from two classes of linear codes
The interplay between coding theory and -designs has attracted a lot of
attention for both directions. It is well known that the supports of all
codewords with a fixed weight in a code may hold a -design. In this paper,
by determining the weight distributions of two classes of linear codes, we
derive infinite families of -designs from the supports of codewords with a
fixed weight in these codes, and explicitly obtain their parameters
Infinite families of -designs from two classes of binary cyclic codes with three nonzeros
Combinatorial -designs have been an interesting topic in combinatorics for
decades. It is a basic fact that the codewords of a fixed weight in a code may
hold a -design. Till now only a small amount of work on constructing
-designs from codes has been done. In this paper, we determine the weight
distributions of two classes of cyclic codes: one related to the triple-error
correcting binary BCH codes, and the other related to the cyclic codes with
parameters satisfying the generalized Kasami case, respectively. We then obtain
infinite families of -designs from these codes by proving that they are both
affine-invariant codes, and explicitly determine their parameters. In
particular, the codes derived from the dual of binary BCH codes hold five
-designs when .Comment: arXiv admin note: text overlap with arXiv:1903.0745
Infinite families of -designs from a class of cyclic codes with two non-zeros
Combinatorial -designs have wide applications in coding theory,
cryptography, communications and statistics. It is well known that the supports
of all codewords with a fixed weight in a code may give a -design. In this
paper, we first determine the weight distribution of a class of linear codes
derived from the dual of extended cyclic code with two non-zeros. We then
obtain infinite families of -designs and explicitly compute their parameters
from the supports of all the codewords with a fixed weight in the codes. By
simple counting argument, we obtain exponentially many -designs.Comment: arXiv admin note: substantial text overlap with arXiv:1903.0745
Steiner systems for supported by a family of extended cyclic codes
In [C. Ding, An infinite family of Steiner systems from cyclic
codes, {\em J. Combin. Des.} 26 (2018), no.3, 126--144], Ding constructed a
family of Steiner systems for all from a
family of extended cyclic codes. The objective of this paper is to present a
family of Steiner systems for all supported
by a family of extended cyclic codes. The main result of this paper complements
the previous work of Ding, and the results in the two papers will show that
there exists a binary extended cyclic code that can support a Steiner system
for all even . This paper also determines the parameters
of other -designs supported by this family of extended cyclic codes
Partial Geometric Designs from Group Actions
In this paper, using group actions, we introduce a new method for
constructing partial geometric designs (sometimes referred to as
-designs). Using this new method, we construct several infinite
families of partial geometric designs by investigating the actions of various
linear groups of degree two on certain subsets of .
Moreover, by computing the stabilizers of such subsets in various linear groups
of degree two, we are also able to construct a new infinite family of balanced
incomplete block designs
Infinite families of 3-designs from APN functions
Combinatorial -designs have nice applications in coding theory, finite
geometries and several engineering areas. The objective of this paper is to
study how to obtain -designs with -transitive permutation groups. The
incidence structure formed by the orbits of a base block under the action of
the general affine groups, which are -transitive, is considered. A
characterization of such incidence structure to be a -design is presented,
and a sufficient condition for the stabilizer of a base block to be trivial is
given. With these general results, infinite families of -designs are
constructed by employing APN functions. Some -designs presented in this
paper give rise to self-dual binary codes or linear codes with optimal or best
parameters known. Several conjectures on -designs and binary codes are also
presented.Comment: 25 page
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