PhD Thesis Defense
Linear codes with Constrained Generator Matrices
Designing good error correcting codes whose generator matrix has a support constraint, i.e., one for which only certain entries of the generator matrix are allowed to be nonzero, has found many recent applications, including in distributed coding and storage, linear network coding, multiple access networks, and weakly secure data exchange. The dual problem, where the parity check matrix has a support constraint, comes up in the design of locally repairable codes. The central problem here is to design codes with the largest possible minimum distance, subject to the given support constraint on the generator matrix. When the distance metric is the Hamming distance, the codes of interest are Reed-Solomon codes, for which case, the problem was formulated as the "GM-MDS conjecture". In the rank metric case, the same problem can be considered for Gabidulin codes. This thesis provides solutions to these problems and discusses the remaining open problems.
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