On the optimum of Delsarte's linear program

Alex Samorodnitsky*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

Abstract

We are interested in the maximal size A(n, d) of a binary error-correcting code of length n and distance d, or, alternatively, in the best packing of balls of radius (d - 1)/2 in the n-dimensional Hamming space. The best known lower bound on A(n, d) is due to Gilbert and Varshamov and is obtained by a covering argument. The best know upper bound is due to McEliece, Rodemich, Rumsey, and Welch, and is obtained using Delsarte's linear programming approach. It is not known whether this is the best possible bound one can obtain from Delsarte's linear program. We show that the optimal upper bound obtainable from Delsarte's linear program will strictly exceed the Gilbert-Varshamov lower bound. In fact, it will be at least as big as the average of the Gilbert-Varshamov bound and the McEliece, Rodemich, Rumsey, and Welch upper bound. Similar results hold for constant weight binary codes. The average of the Gilbert-Varshamov bound and the McEliece, Rodemich, Rumsey, and Welch upper bound might be the true value of Delsarte's bound. We provide some evidence for this conjecture.

Original languageAmerican English
Pages (from-to)261-287
Number of pages27
JournalJournal of Combinatorial Theory. Series A
Volume96
Issue number2
DOIs
StatePublished - 2001
Externally publishedYes

Fingerprint

Dive into the research topics of 'On the optimum of Delsarte's linear program'. Together they form a unique fingerprint.

Cite this