In addition to their applications in data storage, communications systems, and consumer electronics, 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 analyse 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.

BCH codes have been studied for over fifty years and widely employed in consumer devices, communication systems, and data storage systems. However, the dimension of BCH codes is settled only for a very small number of cases. In this paper, we study the dimensions of BCH codes over finite fields with three types of lengths n, namely n = q m − 1, n = (q m − 1)/(q − 1) and n = q m + 1. For narrow-sense primitive BCH codes with designed distance δ, we investigate their dimensions for δ in the range 1 ≤ δ ≤ q ⌈ m 2 ⌉+1 . For non-narrow sense primitive BCH codes, we provide two general formulas on their dimensions and give the dimensions explicitly in some cases. Furthermore, we settle the minimum distances of some primitive BCH codes. We also explore the dimensions of the BCH codes of lengths n = (q m − 1)/(q − 1) and n = q m + 1 over finite fields.

The interplay between coding theory and t-designs started many years ago. While every t-design yields a linear code over every finite field, the largest t for which an infinite family of t-designs is derived directly from a linear or nonlinear code is t = 3. Sporadic 4-designs and 5-designs were derived from some linear codes of certain parameters. The major objective of this paper is to construct many infinite families of 2-designs and 3-designs from linear codes. The parameters of some known t-designs are also derived. In addition, many conjectured infinite families of 2-designs are also presented.

Let F r be an extension of a finite field F q with r = q m . Let each g i be of order n i in F * r and gcd(n i , n j ) = 1 for 1 ≤ i = j ≤ u. We define a cyclic code over F q by C (q,m,n1,n2,...,nu)

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Historically, LCD cyclic codes were referred to as reversible cyclic codes, which had application in data storage. Due to a newly discovered application in cryptography, there has been renewed interest on 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. As a byproduct, the parameters of some primitive BCH codes are also obtained.

In this paper, we study LCD BCH codes over the finite field GF(q) with two types of lengths n, where n = q l + 1 and n = (q l + 1)/(q + 1). Several classes of LCD BCH codes are given and their parameters are determined or bounded by exploring the cyclotomic cosets modulo n. For n = q l + 1, we determine the dimensions of the codes with designed distance δ, where q l+1 2 + 1 ≤ δ ≤ q l+3 2 + 1. For n = (q l + 1)/(q + 1), the dimensions of the codes with designed distance δ are presented, where 2 ≤ δ ≤ q l−1 2 + 1.