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：针对$[n,k]_q$线性码的（汉明）权重集的最大规模问题近期被提出。达到已建立上界的码称为最大权重谱（MWS）码。与$\mathbb{F}_q^n$具有相同权重集的$[n,k]_q$码称为全权重谱（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$M(H,k,q)$的确定仍是开放问题。本研究首先将讨论推广至一般坐标加权函数，进而具体到Lee权重和类曼哈顿权重。在一般情况下，我们给出了存在FWS码的$n$的界，以及存在MWS码的$n$的界。当将问题特化至Lee或曼哈顿框架时，我们能完全确定FWS码的参数。与汉明情形类似，我们给出了$M(\mathcal{L},k,q)$（Lee MWS码的最小长度）的上界，并将$M(\mathcal{L},k,q)$的确定作为开放问题。另一方面，对于曼哈顿权重，我们完全确定了MWS码的参数。