MWS and FWS Codes for Coordinate-Wise Weight Functions

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.