The homogeneous metric can be viewed as a natural extension of the Hamming metric to finite chain rings. It distinguishes between three types of elements: zero, non-zero elements in the socle, and elements outside the socle. Since the Singleton bound is one of the most fundamental and widely studied bounds in classical coding theory, we investigate its analogue for codes over finite chain rings equipped with the homogeneous metric. We provide a complete characterization of Maximum Homogeneous Distance (MHD) codes, showing that MHD codes coincide with lifted MDS codes and are contained within the socle at low rank. Exceptions arise from exceptional MDS codes or single-parity-check codes. We then shift our focus to the Plotkin-type bound in the homogeneous metric and close an existing gap in the theory of constant homogeneous-weight codes by identifying those of minimal length.

翻译：齐次度量可视为汉明度量到有限链环的自然推广。它区分三种元素类型：零元、socle中的非零元以及socle外的元素。由于Singleton界是经典编码理论中最基础且研究最广泛的界之一，我们研究了它在配备齐次度量的有限链环上码的类似形式。我们给出了最大齐次距离（MHD）码的完整刻画，证明MHD码与提升MDS码等价，且在低秩情况下包含于socle中。例外情形源于特殊MDS码或单奇偶校验码。随后，我们将关注点转向齐次度量下的Plotkin型界，并通过识别最小长度的常齐次重量码，填补了该理论中存在的空白。