Linear codes have widespread applications in data storage systems. There are two major contributions in this paper. We first propose infinite families of optimal or distance-optimal linear codes over
...F
p
constructed from projective spaces. Moreover, a necessary and sufficient condition for such linear codes to be Griesmer codes is presented. Secondly, as an application in data storage systems, we investigate the locality of the linear codes constructed. Furthermore, we show that these linear codes are alphabet-optimal locally repairable codes with locality 2.
Codebooks with low-coherence have wide utilization in many fields, such as direct spread code division multiple access communications, compressed sensing and so on. There are two major ingredients in ...this paper. The first is to present a new character sum, the hyper Eisenstein sum and study the properties of this character sum. As an application, the second ingredient is to propose two constructions of codebooks with the hyper Eisenstein sum. The codebooks generated by these constructions asymptotically meet the Welch bound. The parameters of these codebooks are new.
Locally recoverable codes play a crucial role in distributed storage systems. Many studies have only focused on the constructions of optimal locally recoverable codes with regard to the Singleton ...bound. The aim of this paper is to construct optimal binary locally recoverable codes meeting the alphabet-dependent bound. Using a general framework for linear codes associated to a set, we provide a new approach to constructing binary locally recoverable codes with locality 2. We turn the problem of designing optimal binary locally recoverable codes into constructing a suitable set. Several constructions of optimal binary locally recoverable codes are proposed by this new method. Finally, we propose constructions of optimal binary locally recoverable codes with locality 2 and locality parameters <inline-formula> <tex-math notation="LaTeX">(\text {r},\delta) </tex-math></inline-formula> by Griesmer codes.
Linear codes with large minimum distances perform well in error and erasure corrections. Constructing such linear codes is a main topic in coding theory. In this paper, we propose four families of ...linear codes which are optimal or distance-optimal with respect to the Griesmer bound. Using the theory of characters over finite fields, we determine the weight distribution of these linear codes. The results show that these linear codes are two-weight codes. Finally, we analyse the locality of these linear codes and present three families of distance-optimal binary locally repairable code with locality 2 or 3.
In this paper, a family of new entanglement-assisted quantum error-correcting codes (EAQECCs) is constructed from constacyclic codes with length
n
=
q
2
+
1
a
by giving a new method to select ...defining sets, where
a
=
m
2
+
1
,
m
>
2
is even and
q
is an odd prime power with
a
∣
(
q
±
m
)
. It is worth pointing out that those EAQECCs are entanglement-assisted quantum maximum distance separable (EAQMDS) codes when
d
≤
n
+
2
2
. All of them are new in the sense that their parameters are not covered by the previously known ones. Moreover, they have minimum distance larger than
q
+
1
. Compared with the codes with the same length listed in Table 1, our codes have larger minimum distance.
Hulls of linear codes from simplex codes Xu, Guangkui; Luo, Gaojun; Cao, Xiwang ...
Designs, codes, and cryptography,
04/2024, Volume:
92, Issue:
4
Journal Article
Peer reviewed
The hull of a linear code plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear ...code. Regarding the quantum error correction, linear codes with determined hull are used to construct quantum codes. In this paper, we focus on the hull of Simplex codes and punctured Simplex codes. We firstly study the properties of the matrix produced by the column vectors of a projective space and determine the Euclidean and Hermitian hull of punctured Simplex codes completely. Secondly, we investigate the Euclidean and Hermitian hull of several classes of linear codes from Simplex codes.
Linear codes play a key role in widespread applications. In this paper, we propose three new constructions of linear codes. We give some sufficient conditions for the constructed linear codes to be ...optimal or distance-optimal in terms of the Griesmer bound. Three classes of distance-optimal linear codes with new parameters are presented. Under some constraints, we show that some of the presented linear codes have few weights.
Codebooks with low-coherence have wide applications in many fields such as direct spread code division multiple access communications, compressed sensing, signal processing and so on. In this paper, ...we propose two constructions of complex codebooks from the operations of certain sets. The complex codebooks produced by these constructions are proved to be asymptotically optimal with respect to the Welch bound. In addition, the parameters of the complex codebooks presented in this paper are new and flexible in some cases.
Partial spreads are important in finite geometry and can be used to construct linear codes. From the results in (Des. Codes Cryptogr.
90
, 1–15, 2022) by Xia Li, Qin Yue and Deng Tang, we know that ...if the number of the elements in a partial spread is “big enough”, then the corresponding linear code is minimal. This paper used the sufficient condition in (IEEE Trans. Inf. Theory
44
(5), 2010–2017, 1998) to prove the minimality of such linear codes. In the present paper, we use the geometric approach to study the minimality of linear codes constructed from partial spreads in all cases.
Minimal linear codes have received much attention in the past decades due to their important applications in secret sharing schemes and secure two-party computation, etc. Recently, several classes of ...minimal linear codes with
w
min
/
w
max
≤
(
p
−
1
)
/
p
have been discovered, where
w
min
and
w
max
respectively denote the minimum and maximum nonzero weights in a code. In this paper, we investigate the minimality of a class of
p
-ary linear codes and obtain some sufficient conditions for this kind of linear codes to be minimal, which is a generalization of the recent results given by Xu et al. (Finite Fields Appl.
65
,101688,
32
). This allows us to construct new minimal linear codes with
w
min
/
w
max
≤
(
p
−
1
)
/
p
from weakly regular bent functions for the first time. The parameters of minimal linear codes presented in this paper are different from those known in the literature.
