In this paper, we introduce the notion of Jacobi polynomials with multiple reference vectors of a code, and give the MacWilliams type identity for it. Moreover, we derive a formula to obtain the Jacobi polynomials using the Aronhold polarization operator. Finally, we describe some facts obtained from Type III and Type IV codes that interpret the relation between the Jacobi polynomials and designs.

翻译：在本文中,我们引入了具有多种参考矢量代码的雅各比多面体的概念,并给出了该代码的MacWilliams类型身份。此外,我们用亚伦分点极化操作员获取雅各比多面体的公式。最后,我们描述了从解释雅各比多面体和设计之间关系的第三类和第四类代码中获得的一些事实。