The Covering Radius of the Third-Order Reed-Muller Code RM(3,7) is 20

TL;DR

This paper determines the exact covering radius of RM(3,7) as 20, by classifying Boolean functions and proving that higher nonlinearities are impossible, combining theoretical analysis with computational methods.

Contribution

It precisely establishes the covering radius of RM(3,7) as 20, resolving a previously uncertain range, and introduces a classification approach for Boolean functions based on quotient space analysis.

Findings

The covering radius of RM(3,7) is exactly 20.

62 of 66 function types cannot have third-order nonlinearity 21.

Functions of the remaining types can be transformed and shown not to reach nonlinearity 21.

Abstract

We prove the covering radius of the third-order Reed-Muller code RM(3,7) is 20, which was previously known to be between 20 and 23 (inclusive). The covering radius of RM(3, 7) is the maximum third-order nonlinearity among all 7-variable Boolean functions. It was known that there exist 7-variable Boolean functions with third-order nonlinearity 20. We prove the third-order nonlinearity cannot achieve 21. According to the classification of the quotient space of RM(6,6)/RM(3,6), we classify all 7-variable Boolean functions into 66 types. Firstly, we prove 62 types (among 66) cannot have third-order nonlinearity 21; Secondly, we prove function of the remaining 4 types can be transformed into a type (6, 10) function, if its third-order nonlinearity is 21; Finally, we transform type (6, 10) functions into a specific form, and prove the functions in that form cannot achieve third-order nonlinearity 21 (with the assistance of computers). By the way, we prove that the affine transformation group over any finite field can be generated by two elements.