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Linear programming bounds for codes via a covering argument. (English) Zbl 1173.90475
Summary: We 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 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.

MSC:
90C05 Linear programming
94B99 Theory of error-correcting codes and error-detecting codes
65T50 Numerical methods for discrete and fast Fourier transforms
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