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：有关 $[n,k]_q美元线性代码的最大重量的堆积问题,最近引入了 $[rn,k]_q] 线性代码。达到既定上限的代码是最大重量谱(MWS) 代码。这些 $(k)_q美元代码,其重量与$[mathbb{F ⁇ qq ⁇ ] 代码相同。 FWS 代码必须“short”,而 MWS 代码则必须“L8” 。对于固定的 $(n,k)_q 美元,其值为$(n,k)_q美元-FWS 代码。对于固定的值为$(n,k) $($-FWS ) 代码是完全确定的,而最小长度为 $(H) 美元(k) 美元(k) 美元(美元) 美元(美元) 美元(美元), 确定一个最小值(xxx) 美元(MWS) 的代码是完全的。