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

On the parameters of extended primitive cyclic codes and the related designs