Self-orthogonal codes are an important type of linear codes since they are very closely related to designs, lattices, and quantum codes. Bose-Chaudhuri-Hocquenghem codes (BCH codes) have various practical applications in communication and storage due to their efficient encoding and decoding algorithms. In this paper, we will focus on the primitive and projective self-orthogonal BCH codes in both Euclidean and Hermitian cases. Our main objective is to characterize primitive and projective Euclidean and Hermitian self-orthogonal BCH codes and investigate their parameters. For the Euclidean case, the primitive and projective self-orthogonal BCH codes are characterized completely by using their designed distances. For the Hermitian case, a sufficient and necessary condition for all primitive BCH codes being self-orthogonal are presented, while the characterizations of projective Hermitian self-orthogonal BCH codes are obtained in some cases. Moreover, the dimensions of some Euclidean and Hermitian self-orthogonal BCH codes are determined explicitly and lower bounds on their minimum distances are given.

The (Schur) squares of linear codes are an interesting research topic in coding theory, and they have important applications in cryptography. Linear complementary dual codes (LCD codes) have been widely applied in data storage, communication systems, consumer electronics, and cryptography. Given these exciting applications of squares and LCD codes, we mainly focus on the squares of LCD cyclic codes in this paper. It will be proved that the square of an LCD cyclic code is still an LCD cyclic code. As a subclass of cyclic codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes) have explicit defining sets that include consecutive integers, which gives an advantage of analyzing the parameters of BCH codes and their related codes. We will investigate the squares$\mathcal {C}^{2}(t)$and$\mathcal {C}^{2}(t)^{c}$of the primitive LCD BCH codes$\mathcal {C}(t)$and their complements$\mathcal {C}(t)^{c}$, respectively, where$\mathcal {C}(t)=\mathcal {C}_{(q,q^{m}-1,2t,-t+1)}$is the BCH code of length$q^{m}-1$over$\mathbb{F}_{q}$with designed distance$2t$. Two sufficient and necessary conditions to guarantee that$\mathcal {C}^{2}(t) \ne \Bbb \{\textbf {0}\}$and$\mathcal {C}^{2}(t)^{c} \ne \mathbb{F}_{q}^{n}$are proposed by giving restrictions on designed distances. Furthermore, the dimensions and lower bounds on minimum distances of$\mathcal {C}^{2}(t)$and$\mathcal {C}^{2}(t)^{c}$are presented in some cases. The parameters of the squares of the complements of the Melas codes$M(q,m)$are also investigated.

Studying the Schur square of a linear code is an important research topic in coding theory. Schur squares have important applications in cryptography and private information retrieval schemes, notably in secure multiparty computing or designing bilinear multiplication algorithms in finite extensions of finite fields through the notion of supercodes. Thanks to their exciting applications in cryptography, squares and powers of several linear codes have been investigated. In this paper, we will focus on the Schur square of a relevant well-known subclass of cyclic codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes), which have wide applications in communication and storage systems and benefit from explicit defining sets that include consecutive integers, which gives the advantage of analyzing the parameters of BCH codes and their complements. Our main objective is to investigate the parameters of the squares of primitive narrow-sense BCH codes$\mathcal C(\delta)$and their complements$\mathcal C(\delta)^{c}$. We will present two sufficient and necessary conditions to guarantee that$\mathcal C^{2}(\delta) \ne \Bbb F_{q}^{n}$and$\mathcal C^{2}(\delta)^{c} \ne \Bbb F_{q}^{n}$by giving restrictions on designed distance$\delta $, where$2 \le \delta \le n$. Based on these two characterizations, the dimensions and minimum distances of$\mathcal C^{2}(\delta)$and$\mathcal C^{2}(\delta)^{c}$are investigated in some cases. The dimensions of these squares are determined explicitly, and lower bounds on the minimum distance are given.