Liu et al. (IEEE Trans Inf Theory 68:3096–3107, 2022) investigated a class of BCH codes \(\mathcal {C}_{(q,q+1,\delta ,1)}\) with \(q=\delta ^m\) a prime power and proved that the set \(\mathcal {B}_{\delta +1}\) of supports of the minimum weight codewords supports a Steiner system \({{\text {S}}}(3,\delta +1,q+1)\). In this paper, we give an equivalent formulation of \(\mathcal {B}_{\delta +1}\) in terms of elementary symmetric polynomials and then construct a number of mutually disjoint Steiner systems S\((3,\delta +1,\delta ^m+1)\) when m is even and a number of mutually disjoint G-designs G\(\big ({\frac{\delta ^m+1}{\delta +1}},\delta +1,\delta +1,3\big )\) when m is odd. In particular, the existence of three mutually disjoint Steiner systems \({{\text {S}}}(3,5,4^m+1)\) or three mutually disjoint G-designs G\(\big ({\frac{4^m+1}{5}},5,5,3\big )\) is established.

刘等人。(IEEE Trans Inf Theory 68:3096–3107, 2022) 研究了一类 BCH 码\(\mathcal {C}_{(q,q+1,\delta ,1)}\) ，其中\(q=\delta ^m\)素数幂，并证明最小权码字的支持集\(\mathcal {B}_{\delta +1}\)支持 Steiner 系统\({{\text {S}}} (3,\delta +1,q+1)\)。在本文中，我们用初等对称多项式给出了\(\mathcal {B}_{\delta +1}\)的等价公式，然后构造了多个互不相交的 Steiner 系统 S \((3,\delta +1,\delta ^m+1)\)当m为偶数且多个相互不相交的 G 设计 G \(\big ({\frac{\delta ^m+1}{\delta +1}}, \delta +1,\delta +1,3\big )\)当m为奇数时。特别是，存在三个相互不相交的斯坦纳系统\({{\text {S}}}(3,5,4^m+1)\)或三个相互不相交的 G 设计 G \(\big ({\ frac{4^m+1}{5}},5,5,3\big )\)成立。