The notion of T-design in a symmetric association scheme was introduced by Delsarte. A harmonic index t-design is the particular case in which the index set T consists of a single index \(\{t\}\). Zhu et al. studied a harmonic index t-design in the binary Hamming scheme and gave a Fisher type lower bound on the cardinality. Also they defined the notion of a tight harmonic index design using the bound, and considered the classification problem. In this paper, we extend their results to the general Hamming scheme and improve their Fisher type lower bound slightly. Also using the improved Fisher type lower bound, we redefine the notion of a tight harmonic index design and consider the classification problem. Furthermore, we give a natural characterization for a harmonic index t-design and an analogue in the Hamming scheme for the construction of spherical harmonic index t-designs which was given by Bannai–Okuda–Tagami.

Delsarte 引入了对称关联方案中的T设计概念。谐波索引t -design 是索引集T由单个索引\(\{t\}\)组成的特殊情况。朱等人。研究了一个谐波指数t-在二元汉明方案中设计，并给出了基数的 Fisher 类型下界。他们还使用界限定义了紧谐波指数设计的概念，并考虑了分类问题。在本文中，我们将他们的结果扩展到一般的汉明方案，并稍微改进了他们的 Fisher 类型下界。同样使用改进的 Fisher 类型下界，我们重新定义了紧谐波指数设计的概念并考虑了分类问题。此外，我们给出了谐波指数t -design的自然表征，以及由 Bannai-Okuda-Tagami 给出的用于构建球谐指数t -design的 Hamming 方案中的类似物。