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 <inline formula> <tex math notation="LaTeX"\mathcal {C}_{(r,r^{m} 1,\delta,b) /tex math>/inline formula> be the primitive BCH code over <inline formula> <tex math notation="LaTeX"\mathbb {F}_{r} /tex math>/inline formula> of length <inline formula> <tex math notation="LaTeX"$r^{m} 1$ /tex math>/inline formula> with designed distance <inline formula> <tex math notation="LaTeX"\delta /tex math>/inline formula> where <inline formula> <tex math notation="LaTeX"\mathbb {F}_{r} /tex math>/inline formula> is the finite field of order <inline formula> <tex math notation="LaTeX"$r$ /tex math>/inline formula> 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 self orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code <inline formula> <tex math notation="LaTeX"\mathcal {C}_{(r,r^{2} 1,\delta,1) /tex math>/inline formula> and its hull in both Hermitian and Euclidean cases for <inline formula> <tex math notation="LaTeX"$2 \le \delta \le r^{2} 1$ /tex math>/inline formula> We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.
IEEE Transactions on Information Theory