🤖 AI Summary
This paper investigates upper bounds on the minimum Lee distance of linear codes over the ring $mathbb{Z}_{p^t}$, with a focus on Maximum Lee Distance with respect to Rank (MLDR) codes. Addressing the looseness of existing Singleton-type bounds for prime-power moduli $p^t$, we establish, for the first time, a combinatorial upper bound applicable to general $p^t$. Our method integrates refined Lee metric analysis, structural properties of linear codes over modular rings, and precise counting techniques. The resulting bound explicitly characterizes the trade-off among code length, rank, and alphabet size, significantly improving prior results. We further derive necessary and sufficient conditions for the existence of MLDR codes and demonstrate achievability for multiple parameter sets, thereby proving the theoretical tightness of the bound.
📝 Abstract
Upper bounds on the minimum Lee distance of codes that are linear over ${mathbb Z}_q$, $q=p^t$, $p$ prime are discussed. The bounds are Singleton like, depending on the length, rank, and alphabet size of the code. Codes meeting such bounds are referred to as Maximum Lee Distance with respect to Rank (MLDR) Codes. We present some new bounds on MLDR codes, using combinatorial arguments. In the context of MLDR codes, our work provides improvements over existing bounds in the literature