In this work, we determine new linear equations for the weight distribution of linear codes over finite chain rings. The identities are determined by counting the number of some special submatrices of the parity-check matrix of the code. Thanks to these relations we are able to compute the full weight distribution of codes with small Singleton defects, such as MDS, MDR and AMDR codes.

翻译：在这项工作中,我们决定了线性代码在有限链环上的重量分布的新的线性方程。 确定身份的方法是计算代码对等检查矩阵中某些特殊次矩阵的数量。 由于这些关系,我们得以计算出单质小缺陷( 如MDS、 MDR 和 AMDR ) 代码的全部重量分布 。