ECCC-Report TR07-033https://eccc.weizmann.ac.il/report/2007/033Comments and Revisions published for TR07-033en-usTue, 03 Apr 2007 13:33:56 +0300
Paper TR07-033
| Linear programming bounds for codes via a covering argument |
Michael Navon,
Alex Samorodnitsky
https://eccc.weizmann.ac.il/report/2007/033We recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and designs via a covering argument. It is possible to show, interpreting the following notions appropriately, that if a code has a large distance, then its dual has a small covering radius and, therefore, is large. This implies the original code to be small.
We also point out (in conjunction with further work) that this bound is a natural isoperimetric constant of the Hamming cube, related to its Faber-Krahn minima.
While our approach belongs to the general framework of Delsarte's linear programming method, its main technical ingredient is Fourier duality for the Hamming cube. In particular, we do not deal directly with Delsarte's linear program or orthogonal polynomial theory.
Tue, 03 Apr 2007 13:33:56 +0300https://eccc.weizmann.ac.il/report/2007/033