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A linear [n, k]q code C is said to be a full weight spectrum (FWS) code if there exist codewords of each weight less than or equal to n. In this brief communication we determine necessary and sufficient conditions for the existence of linear [n, k]q full weight spectrum (FWS) codes. Central to our approach is the geometric view of linear codes, whereby columns of a generator matrix correspond to points in P G(k − 1, q).Recently, Shi et. al. [10,9] studied a combinatorial problem concerning the maximum number L(k, q) of distinct weights a linear code of dimension k over GF (q) may realize. Obviously L(k, q) ≤ q k −1 q−1 . In [10], this bound is shown to be sharp for binary codes, and for all q-ary codes of dimension k = 2. We note that in 2015, Haily and Harzalla [7] also established the existence of binary codes meeting this bound. Shi et. al. went on to conjecture that the bound is sharp for all q and k. This conjecture was proved correct in [1]. Codes meeting this bound are called maximum weight spectrum (MWS) codes.A further refinement was also investigated in [10] by the introduction of the function L(n, k, q), denoting the maximum number of non-zero weights an [n, k] q code may have. They observed that an immediate upper bound is L(n, k, q) ≤ n. In this short communication we establish necessary and sufficient conditions for the existence of codes meeting this bound. Such codes will be said to be full weight spectrum (FWS) codes. MSC(2010): Primary: 94B05 ; Secondary: 94B65 and 94B25.

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An (n, k) q -MDS code C over an alphabet A (of size q) is a collection of q k ntuples over A such that no two words of C agree in as many as k coordinate positions. It follows that n ≤ q + k − 1. By elementary combinatorial means we show that every (6, 3) 4 -MDS code, linear or not, turns out to be a linear (6, 3) 4 -MDS code or else a code equivalent to a linear code with these parameters. It follows that every (5, 3) 4 -MDS code over A must also be equivalent to linear.