On the Squares of LCD Cyclic Codes and Their Complements: Study of Several Families and Analyzing Their Parameters

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 C-2(t) and C-2(t)(c) of the primitive LCD BCH codes C(t) and their complements C(t)(c) , respectively, where C(t)=C-(q,C-qm-1,C-2t,C--t+1) is the BCH code of length q(m)-1 over F-q with designed distance 2t . Two sufficient and necessary conditions to guarantee that C-2(t)not equal{0} and C-2(t)F-c not equal(q)n are proposed by giving restrictions on designed distances. Furthermore, the dimensions and lower bounds on minimum distances of C-2(t) and 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.