A combinatorial problem concerning the maximum size of the (hamming) weight set of an $[n,k]_q$ linear code was recently introduced. Codes attaining the established upper bound are the Maximum Weight Spectrum (MWS) codes. Those $[n,k]_q $ codes with the same weight set as $ \mathbb{F}_q^n $ are called Full Weight Spectrum (FWS) codes. FWS codes are necessarily ``short", whereas MWS codes are necessarily ``long". For fixed $ k,q $ the values of $ n $ for which an $ [n,k]_q $-FWS code exists are completely determined, but the determination of the minimum length $ M(H,k,q) $ of an $ [n,k]_q $-MWS code remains an open problem. The current work broadens discussion first to general coordinate-wise weight functions, and then specifically to the Lee weight and a Manhattan like weight. In the general case we provide bounds on $ n $ for which an FWS code exists, and bounds on $ n $ for which an MWS code exists. When specializing to the Lee or to the Manhattan setting we are able to completely determine the parameters of FWS codes. As with the Hamming case, we are able to provide an upper bound on $ M(\mathcal{L},k,q) $ (the minimum length of Lee MWS codes), and pose the determination of $ M(\mathcal{L},k,q) $ as an open problem. On the other hand, with respect to the Manhattan weight we completely determine the parameters of MWS codes.

翻译：最近引入了一个关于$[n,k]_q$线性码（汉明）重量集最大规模的组合问题。达到既定上界的码称为最大重量谱（MWS）码。那些与$\mathbb{F}_q^n$具有相同重量集的$[n,k]_q$码称为全重量谱（FWS）码。FWS码必然是“短码”，而MWS码必然是“长码”。对于固定的$k,q$，存在$[n,k]_q$-FWS码的$n$值已被完全确定，但$[n,k]_q$-MWS码的最小长度$M(H,k,q)$的确定仍是一个未解决问题。当前工作首先将讨论拓展至一般坐标方向加权函数，进而具体针对李权重和类曼哈顿权重展开。在一般情况下，我们给出了存在FWS码的$n$值边界以及存在MWS码的$n$值边界。当特化至李权重或曼哈顿权重设定时，我们能够完全确定FWS码的参数。与汉明情形类似，我们给出了$M(\mathcal{L},k,q)$（李MWS码的最小长度）的上界，并将$M(\mathcal{L},k,q)$的确定作为未解决问题提出。另一方面，针对曼哈顿权重，我们完全确定了MWS码的参数。