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介绍了对称关联方案中的T-设计概念。谐波指标t -design是一种特殊情况，其中指标集T由单个指标\（\ {t \} \）组成。朱等。研究了谐波指数t-在二进制Hamming方案中进行设计，并给出了基数的Fisher类型下界。他们还使用边界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，我们将其结果扩展到一般的汉明方案，并稍微改善其Fisher类型下界。同样使用改进的Fisher类型下界，我们重新定义了紧调和指数设计的概念，并考虑了分类问题。此外，我们给出了谐波指数t-设计的自然表征，并给出了汉明方案中由班纳伊-奥克达-塔塔米给出的球形谐波指数t-设计的类似物。