GHW-based Necessary and Sufficient Conditions for Support Constrained Generator Matrices

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.