On Hulls of Some Primitive BCH Codes and Self-Orthogonal Codes

Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let C-(r,C-rm-1,C-delta,C-b) be the primitive BCH code over F-r of length r(m) - 1 with designed distance delta, where F-r is the finite field of order r(m) In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian selforthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code C-(r,C- r2-1,C-delta,C-1) and its hull in both Hermitian and Euclidean cases for 2 <= delta <= r(2) - 1. We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.

Automated summary

This paper investigates self-orthogonal codes through the study of the hulls of linear codes, obtaining parameters for Euclidean and Hermitian self-orthogonal codes based on primitive BCH codes. It also develops several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls, and proposes Hermitian self-orthogonal codes through the hulls of BCH codes. Additionally, the dimensions of codes and their hulls in both Hermitian and Euclidean cases are determined for specific delta values, along with conditions for achieving the largest dimension in the hull.