  Support constrained generator matrix for a linear code has been an active
topic in recent years. The necessary and sufficient condition for the existence
of MDS codes over small fields with support constrained generator matrices were
conjectured by Dau, Song, Yuen in 2014. This GM-MDS conjecture was proved
independently by Lovett and Yildiz-Hassibi in 2018. In this paper we propose
the necessary and sufficient conditions for support constrained generator
matrices of general linear codes based on the generalized Hamming weights. It
is proved that the direct generalization of the GM-MDS conjecture for $2$-MDS
codes and algebraic geometry codes is not true over arbitrary fields. We
propose and prove a GHW-based sufficient condition for support constrained
matrices of general linear codes. This is the first sufficient condition for
the existence of support constrained generator matrices for general linear
codes over arbitrary finite fields. Moreover a weaker GHW-based sufficient
condition for support constrained generator matrices are given for the binary
simplex code and the first order $q$-ary Reed-Muller codes.
