The number-theoretic codes are a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts' codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo $r$.

翻译：数字理论代码是由单一或多重一致性定义的代码类别。 这些代码主要用于纠正插入和删除错误,以及纠正不对称错误。 本文为数字理论代码的完整重量计算器的概括化提供了一个公式。 这个公式允许我们计算数字理论代码的重量计算器和基点。 作为特例,本文件提供了非二进制 Tenengolts 代码的含重计算器和基点, 校正单插入或删除。 此外, 我们显示, 公式在整数摩杜洛 $ 的环上推断线性代码的MacWilliams 身份 。