Problem
Research questions and friction points this paper is trying to address.
homogeneous metric
Singleton bound
finite chain rings
constant homogeneous-weight codes
Plotkin-type bound
On Optimal Homogeneous-Metric Codes
🤖 AI Summary
This study investigates coding theory over finite chain rings under the homogeneous metric, focusing on analogues of the Singleton bound and Plotkin-type bounds for constant homogeneous-weight codes, as well as associated minimum length problems. By leveraging algebraic coding theory and structural properties of finite chain rings, the work provides the first complete characterization of maximum homogeneous distance (MHD) codes, proving their equivalence to lifted MDS codes. In the low-rank case, it further reveals that such codes are contained in the socle of the ring. Additionally, the minimum length required for constant homogeneous-weight codes to attain the Plotkin-type bound is precisely determined. These results fill a critical gap in the bound theory under the homogeneous metric and significantly advance the generalization of classical coding bounds within this metric framework.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
homogeneous metric
Maximum Homogeneous Distance codes
finite chain rings
lifted MDS codes
Plotkin-type bound
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