Support Constrained Generator Matrices and the Generalized Hamming Weights

Support constrained generator matrices for linear codes have been found applications in multiple access networks and weakly secure document exchange. The necessary and sufficient conditions for the existence of Reed-Solomon codes with support constrained generator matrices were conjectured by Dau, Song, Yuen and Hassibi. This conjecture is called the GM-MDS conjecture and finally proved recently in independent works of Lovett and Yildiz-Hassibi. From their conjecture support constrained generator matrices for MDS codes are existent over linear size small fields. In this paper we propose a natural generalized conjecture for the support constrained matrices based on the generalized Hamming weights (SCGM-GHW conjecture). The GM-MDS conjecture can be thought as a very special case of our SCGM-GHW conjecture for linear $1$-MDS codes. We investigate the support constrained generator matrices for some linear codes such as $2$-MDS codes, first order Reed-Muller codes, algebraic-geometric codes from elliptic curves from the view of our SCGM-GHW conjecture. In particular the direct generalization of the GM-MDS conjecture about $1$-MDS codes to $2$-MDS codes is not true. For linear $2$-MDS codes only cardinality-based constraints on subset systems are not sufficient for the purpose that these subsets are in the zero coordinate position sets of rows in generator matrices.