Optimum Linear Codes With Support-Constrained Generator Matrices Over Small Fields

• Published in: IEEE transactions on information theory Vol. 65; no. 12; pp. 7868 - 7875
• Main Authors: Yildiz, Hikmet, Hassibi, Babak
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.12.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Codes; Constraints; distributed codes; Generators; Linear codes; Mathematical analysis; Matrix methods; minimum distance; Optimization; Reed–Solomon codes; Relays; Silicon; Upper bound
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2019.2932663

We 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 <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>, and maximum minimum distance <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> (over all generator matrices with the given support), then an optimal code exists for any field size <inline-formula> <tex-math notation="LaTeX">q\geq 2n-d </tex-math></inline-formula>. As a by-product of this result, we settle the GM-MDS conjecture in the affirmative.