论文标题
MWS和FWS代码，用于坐标的重量功能
MWS and FWS Codes for Coordinate-Wise Weight Functions
论文作者
论文摘要
最近引入了$ [n，k] _q $线性代码的（Hamming）重量集的最大尺寸的组合问题。达到已建立上限的代码是最大重量谱（MWS）代码。这些$ [n，k] _q $代码与$ \ mathbb {f} _q^n $相同的重量集称为全重量谱（FWS）代码。 FWS代码一定是``简短''，而MWS代码一定是``长期''。对于固定的$ k，q $，完全确定了$ [n，k] _q $ -fws代码的$ n $的值，但确定最小长度$ m（h，k，q）$ a $ [n，k] _q $ -mws代码的确定仍然是一个公开问题。当前的工作首先将讨论扩展到一般坐标的重量功能，然后专门针对Lee的体重和曼哈顿的体重。在一般情况下，我们为存在FWS代码的$ N $提供界限，并在其$ n $上限制了MWS代码的存在。专门用于LEE或曼哈顿设置时，我们可以完全确定FWS代码的参数。与锤子案例一样，我们能够在$ m（\ Mathcal {l}，k，q）$（Lee MWS代码的最小长度）上提供上限，并提出$ M（\ Mathcal {l}，K，Q）的确定。另一方面，关于曼哈顿的重量，我们完全确定了MWS代码的参数。
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.
