The relative hull of a linear codeC1with respect to another linear codeC2is defined as the intersection ofC1and the dual ofC2, i.e.,C1∩C⊥2. Andersonet al. [2] demonstrated that the dimension of the relative hull can be adjusted, either repeatedly increased or decreased by one, until a certain bound is reached by substituting eitherC1orC2with its equivalent code. As a special class of linear codes, BCH codes are both theoretically significant and practically valuable for communication and storage systems due to their good algebraic structures and flexible error-correcting capabilities. This raises an important question: how does the dimension of the relative hull change when C1 and C2 are BCH codes or their equivalent codes? This paper focuses on the relative hull dimensions of narrow-sense primitive BCH codes, building on the insights from [2]. We present several sufficient and necessary conditions based on the designed distances of BCH codes, which ensure that the dimensions of the relative hulls reach the lower or upper bounds for linear codes. Additionally, we study the parameters of the relative hulls of various classes of BCH codes, providing details on their dimensions and developing lower bounds for their minimum distances. Furthermore, we investigate how the dimensions of the relative hulls are affected when the BCH codesC2are replaced by their equivalent codesC′2.

Abstract. Affine-invariant codes are an important class of linear codes, which are extended cyclic codes of length [Formula: see text] invariant under the affine groups acting on [Formula: see text]. These codes are closely related to combinatorics, as they can be applied to construct [Formula: see text]-designs. It is known that the classical Reed–Muller codes and extended primitive narrow-sense Bose–Chaudhuri–Hocquenghem codes are affine-invariant. The objective of this paper is to construct a class of affine-invariant codes [Formula: see text] and investigate the parameters of these codes and their related codes. The dimensions of the codes [Formula: see text] and [Formula: see text] with [Formula: see text] are presented and a recursive formula to compute the dimensions of [Formula: see text] and [Formula: see text] is developed for general [Formula: see text], where [Formula: see text] is the extended code of [Formula: see text]. Meanwhile, lower bounds on minimum distances of [Formula: see text] and [Formula: see text] are also given. Moreover, the parameters and the borders of the dual codes [Formula: see text] are investigated. Two necessary and sufficient conditions for [Formula: see text] being self-orthogonal with [Formula: see text] are developed by employing their borders. In addition, for [Formula: see text] we explore the parameters of the hull of [Formula: see text] and determine its border. It should be pointed out that several affine-invariant self-orthogonal codes will be obtained.

In coding theory, the well-known class of block codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes), form a class of cyclic error-correcting codes constructed using polynomials over a finite field. They are used for various critical practical applications in communication and storage due to their efficient encoding and decoding algorithms. In the past sixty years, significant progress has been made in understanding BCH codes’ dimensions and minimum distances. However, there has been limited research on the minimum distances of the dual codes of BCH codes, making it challenging to determine their actual minimum distances. Therefore, developing accurate lower bounds on the minimum distances of the dual codes of BCH codes is crucial and exciting. In this paper, we primarily use the multiplier technique proposed by Huffman and Pless to investigate the lower bounds on minimum distances of the dual codes$\mathcal {C}_{(q,q^{m}-1,\delta)}^{\perp } $of the primitive narrow-sense BCH codes with designed distance$\delta $. When$q = p^{e}$with$e \ge 2$, we improve the lower bounds on minimum distances of the dual codes$\mathcal {C}_{(q,q^{m}-1,\delta)}^{\perp } $in the ranges$p^{ei}-p^{e-1}+2 \le \delta \le p^{ei+e-1}-p^{e-1}+1$, where$m \ge 2$and$1 \le i \le m-1$. These new lower bounds are much tighter than the previously known bounds in the literature. This technique also applies to the study of binary dual codes$\mathcal {C}_{(2,2^{m}-1,\delta)}^{\perp } $, for which we obtain tight lower bounds for$\delta = 2^{t}$, where$m \ge 5$is odd and$2 \le t \le m-3$is even.

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$\mathcal {C}_{(r,r^{m}-1,\delta,b)}$be the primitive BCH code over$\mathbb {F}_{r}$of length$r^{m}-1$with designed distance$\delta $, where$\mathbb {F}_{r}$is the finite field of order$r$. In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code$\mathcal {C}_{(r,r^{2}-1,\delta,1)}$and its hull in both Hermitian and Euclidean cases for$2 \le \delta \le r^{2}-1$. We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.