Very recently, Heng et al. studied a family of extended primitive cyclic codes. It was shown that the supports of all codewords with any fixed nonzero Hamming weight of this code supporting 2-designs. In this paper, we study this family of extended primitive cyclic codes in more details. The weight distribution is determined. The parameters of the related $2$-designs are also given. Moreover, we prove that the codewords with minimum Hamming weight supporting 3-designs, which gives an affirmative solution to Heng's conjecture.

翻译：近期，Heng等人研究了一类扩展本原循环码。结果表明，该码中所有具有任意固定非零汉明重量的码字支撑集构成2-设计。本文进一步详细研究了这类扩展本原循环码，确定了其重量分布，并给出了相关$2$-设计的参数。此外，我们证明了最小汉明重量码字支撑集构成3-设计，从而肯定地解决了Heng的猜想。