A combinatorial problem concerning the maximum size of the (hamming) weight
set of an [n,k]qβ linear code was recently introduced. Codes attaining the
established upper bound are the Maximum Weight Spectrum (MWS) codes. Those
[n,k]qβ codes with the same weight set as Fqnβ 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]qβ-FWS code exists are completely determined, but the determination
of the minimum length M(H,k,q) of an [n,k]qβ-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(L,k,q) (the minimum length of Lee MWS codes),
and pose the determination of M(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.Comment: 17 page