On the Covering Radius of the Second Order Reed-Muller Code of Length 128

TL;DR

This paper establishes an upper bound of 42 for the covering radius of the second-order Reed-Muller code RM(2,7), advancing understanding of its error-correcting capabilities and characterizing functions achieving this bound.

Contribution

It proves the exact upper bound of 42 for RM(2,7)'s covering radius and provides a necessary and sufficient condition for Boolean functions to reach this nonlinearity.

Findings

Covering radius of RM(2,7) is at most 42.

Characterization of Boolean functions with second-order nonlinearity 42.

Abstract

In 1981, Schatz proved that the covering radius of the binary Reed-Muller code $RM(2,6)$ is 18. For $RM(2,7)$, we only know that its covering radius is between 40 and 44. In this paper, we prove that the covering radius of the binary Reed-Muller code $RM(2,7)$ is at most 42. Moreover, we give a sufficient and necessary condition for Boolean functions of 7-variable to achieve the second-order nonlinearity 42.