Boolean functions have important applications in cryptography and coding theory. Two famous classes of binary codes derived from Boolean functions are the Reed–Muller codes and Kerdock codes. In the ...past two decades, a lot of progress on the study of applications of Boolean functions in coding theory has been made. Two generic constructions of binary linear codes with Boolean functions have been well investigated in the literature. The objective of this paper is twofold. The first is to provide a survey on recent results, and the other is to propose open problems on one of the two generic constructions of binary linear codes with Boolean functions. These open problems are expected to stimulate further research on binary linear codes from Boolean functions.
Linear Codes From Some 2-Designs Ding, C
IEEE transactions on information theory,
2015-June, 2015-6-00, 20150601, Volume:
61, Issue:
6
Journal Article
Peer reviewed
A classical method of constructing a linear code over GF(q) with a t-design is to use the incidence matrix of the t-design as a generator matrix over GF(q) of the code. This approach has been ...extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 2-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 2-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterization of highly nonlinear Boolean functions is presented.
In this paper, a class of two-weight and three-weight linear codes over GF(p) is constructed, and their application in secret sharing is investigated. Some of the linear codes obtained are optimal in ...the sense that they meet certain bounds on linear codes. These codes have applications also in authentication codes, association schemes, and strongly regular graphs, in addition to their applications in consumer electronics, communication and data storage systems.
Parameters of Several Classes of BCH Codes Ding, Cunsheng
IEEE transactions on information theory,
2015-Oct., 2015-10-00, 20151001, Volume:
61, Issue:
10
Journal Article
Peer reviewed
Because of their efficient encoding and decoding algorithms, cyclic codes-an interesting class of linear codes- are widely used in communication systems, storage devices, and consumer electronics. ...BCH codes form a special class of cyclic codes, and are usually among the best cyclic codes. A subclass of good BCH codes is the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The main objective of this paper is to study the dimension and minimum distances of a subclass of the narrow-sense primitive BCH codes with design distance δ = (q - ℓ 0 )q m-ℓ 1 -1 -1 for certain pairs (ℓ 0 , ℓ 1 ), where 0 ≤ ℓ 0 q - 2 and 0 ≤ ℓ 1 m - 1. The parameters of other related classes of BCH codes are also investigated, and some open problems are proposed in this paper.
Binary Linear Codes With Three Weights Ding, Kelan; Ding, Cunsheng
IEEE communications letters,
2014-Nov., 2014-11-00, Volume:
18, Issue:
11
Journal Article
Peer reviewed
In this paper, a class of binary three-weight linear codes is constructed. Their dual codes are also studied. The dual codes are either optimal or almost optimal. These codes have applications in ...authentication codes, secret sharing schemes, and association schemes, in addition to their applications in consumer electronics and communication and data storage systems.
LCD Cyclic Codes Over Finite Fields Li, Chengju; Ding, Cunsheng; Li, Shuxing
IEEE transactions on information theory,
07/2017, Volume:
63, Issue:
7
Journal Article
Peer reviewed
In addition to their applications in data storage, communications systems, and consumer electronics, linear complementary dual (LCD) codes-a class of linear codes-have been employed in cryptography ...recently. LCD cyclic codes were referred to as reversible cyclic codes in the literature. The objective of this paper is to construct several families of reversible cyclic codes over finite fields and analyze their parameters. The LCD cyclic codes presented in this paper have very good parameters in general, and contain many optimal codes. A well rounded treatment of reversible cyclic codes is also given in this paper.
Irreducible cyclic codes have been an interesting subject of study for a long time. Their weight distribution is known in only a few cases. In this paper, the weight distribution of the irreducible ...cyclic codes in a number of other cases is determined. The number of nonzero weights in the codes dealt with in this paper varies between one and four.
This is the first monograph on codebooks and linear codes from difference sets and almost difference sets. It aims at providing a survey of constructions of difference sets and almost difference sets ...as well as an in-depth treatment of codebooks and linear codes from difference sets and almost difference sets. To be self-contained, this monograph covers necessary mathematical foundations and the basics of coding theory. It also contains tables of best BCH codes and best cyclic codes over GF(2) and GF(3) up to length 125 and 79, respectively. This repository of tables can be used to benchmark newly constructed cyclic codes. This monograph is intended to be a reference for postgraduates and researchers who work on combinatorics, or coding theory, or digital communications.
The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in ...irreducible cyclic codes.
Two Families of LCD BCH Codes Li, Shuxing; Li, Chengju; Ding, Cunsheng ...
IEEE transactions on information theory,
09/2017, Volume:
63, Issue:
9
Journal Article
Peer reviewed
Historically, LCD cyclic codes were referred to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed ...interest in LCD codes. In this paper, we explore two special families of LCD cyclic codes, which are both BCH codes. The dimensions and the minimum distances of these LCD BCH codes are investigated.
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