Products of Some Primitive BCH Codes and Their Complements

Schur product was originally proposed in coding theory for algebraic decoding algorithms and widely applied to solve some cryptographic problems in recent years. This shows the great importance of the Schur product in both coding theory and cryptography. As a well-known subclass of cyclic codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes) have wide applications in communication and storage systems. Let C₁ and C₂ be two primitive BCH codes over F_q with designed distances δ_a and δ_b, respectively, where 2 ≤ δ_a, δ_b ≤ n. Let C₁^c and C₂^c be the complements of C₁ and C₂, respectively. This paper aims to investigate the parameters of the products C₁∗C₂ and C₁^c∗C₂^c. We will present some sufficient and necessary conditions to guarantee that C₁∗C₂≠F_qⁿ and C₁^c∗C₂^c≠F_qⁿ by giving restrictions on the designed distances δ_a and δ_b of the two BCH codes, respectively. The dimensions of these products are determined explicitly and lower bounds on the minimum distance are developed in some cases. Some optimal or best known codes are found. Moreover, it should be emphasized that a class of [n, k, d] cyclic codes over Fq with dimension k ≥ n/2 and d ≥ √n are presented.