Optimum Linear Codes With Support-Constrained Generator Matrices Over Small Fields
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AbstractWe consider the problem of designing optimal linear codes (in terms of having the largest minimum distance) subject to a support constraint on the generator matrix. We show that the largest minimum distance can be achieved by a subcode of a Reed-Solomon code of small field size and with the same minimum distance. In particular, if the code has length n , and maximum minimum distance d (over all generator matrices with the given support), then an optimal code exists for any field size q≥ 2n-d. As a by-product of this result, we settle the GM-MDS conjecture in the affirmative.
CitationYildiz, H., & Hassibi, B. (2019). Optimum Linear Codes With Support-Constrained Generator Matrices Over Small Fields. IEEE Transactions on Information Theory, 65(12), 7868–7875. doi:10.1109/tit.2019.2932663
SponsorsThis work was supported in part by the National Science Foundation under Grant CNS-0932428, Grant CCF-1018927, Grant CCF1423663, and Grant CCF-1409204, in part by Qualcomm Inc., in part by the NASAs Jet Propulsion Laboratory through the President and Directors Fund, and in part by the King Abdullah University of Science and Technology