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：-
“坐标加权函数的MWS和FWS代码”
翻译摘要：
该论文最近提出了一个关于（汉明）重量集的最大大小的组合问题，即线性代码$[n，k]_q$的最大权谱（MWS）代码。满足与$\mathbb{F}_q^n$相同权集的代码称为完全权谱（FWS）代码。FWS代码必然是“短”的，而MWS代码必然是“长”的。对于固定的$k，q$值，已完全确定存在$ [n，k]_q $-FWS代码的$ n $值，但确定$ [n，k]_q $ -MWS代码的最小长度$M（H，k，q）$仍然是一个未解决的问题。当前工作首先广泛讨论一般的坐标加权函数，然后具体到Lee重量和类曼哈顿重量。在一般情况下，我们提供了存在FWS代码的$ n $的界限和存在MWS代码的$ n $界限。当专门针对Lee型或类曼哈顿型角度进行阐述时，我们能够完全确定FWS代码的参数。与海明情况相似，我们能够提供$M（\mathcal{L}，k，q）$（Lee MWS代码的最小长度）的上界，并提出$M(\mathcal{L},k,q)$的决定作为一个未解决的问题。另一方面，关于曼哈顿加权，我们完全确定了MWS代码的参数。
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