Support Constrained Generator Matrices and the Generalized Hamming Weights
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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.
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