In this project, we studied the constructions of MDS codes with structurally imposed generator matrices. In particular, our GM-MDS conjecture states that any kxn binary matrix (mask) that does not violate an obvious condition can always be turned into a generator matrix of an MDS code over a finite field of size polynomial in n. A definitive answer to this conjecture remains largely unknown, despite the significant effort of several research groups in the coding community. The conjecture has been confirmed only for k <= 5 and for some other special cases.
[4] H. Dau, W. Song, and C. Yuen, “On simple multiple access networks,” IEEE Journal on Selected Areas in Communications (JSAC), volume 33, number 2, pages 236-249, 2015.
[3] H. Dau, W. Song, and C. Yuen, “Weakly secure MDS codes for simple multiple access network”, IEEE International Symposium on Information Theory (ISIT), pages 1941-1945, 2015.
[2] H. Dau, W. Song, and C. Yuen, "On the Existence of MDS Codes over Small Fields with Constrained Generator Matrices," IEEE International Symposium on Information Theory (ISIT), pages 1787-1791, 2014.
[1] H. Dau, W. Song, Z. Dong, and C. Yuen, "Balanced Sparsest Generator Matrices for MDS Codes," in Proceedings of the IEEE International Symposium on Information Theory (ISIT), pages 1889-1893, 2013.