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