Characterizations of Primitive and Projective Self-Orthogonal BCH Codes and Their Parameters

Self-orthogonal codes are an important type of linear codes since they are very closely related to designs, lat tices, 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.