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 in this code support a 2-design. In this paper, we study this family of extended primitive cyclic codes in more details. The weight distribution is determined and the parameters of the related 2-designs are also given. Moreover, we prove that the minimum weight codewords in this code support a 3-design when \(p=2\), which gives an affirmative answer to Heng’s conjecture.

中文翻译：

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扩展原语循环码的参数及相关设计

最近，Heng 等人。研究了一系列扩展的原始循环码。结果表明，该代码中具有任何固定非零汉明权重的所有码字的支持都支持 2 设计。在本文中，我们更详细地研究了这一系列扩展的原始循环码。确定了权重分布并给出了相关2-设计的参数。此外，我们证明了当\(p=2\)时，该代码中的最小权码字支持3-设计，这对Heng的猜想给出了肯定的答案。