The Weight Distribution of the Third-Order Reed-Muller Code of Length 2048

2026-07-02

AI Summary

This study investigates the weight distribution of the third-order Reed–Muller code RM(3,11) of length 2048 and the covering radii of certain subcodes. By analyzing the 3,691,560 nonzero orbit representatives of ternary Boolean cubic forms under the action of GL(10,2), and leveraging coset weight enumerators together with structural theorems—specifically, that every nondegenerate Boolean cubic form admits a nondegenerate hyperplane restriction except for a unique orbit in odd dimensions—the complete weight distribution of RM(3,11) is determined. Furthermore, the relative covering radius of RM(2,10) in RM(3,10) is precisely improved to 408, and the upper bound on the relative covering radius of RM(6,10) in RM(7,10) is significantly reduced from 50 to 32.

Abstract

We compute the weight distribution of the third-order Reed--Muller code RM(3,11) of length 2048. The weight enumerator is assembled from the coset weight enumerators of f+RM(2,10), evaluated for representatives of all 3691560 nonzero GL(10,2)-orbits of Boolean cubic forms in ten variables. The computation rests on a structural theorem: a nondegenerate Boolean cubic form admits a nondegenerate hyperplane restriction, except for a single orbit in each odd dimension. The same pass determines the second-order nonlinearity of every cubic form: the relative covering radius of RM(2,10) in RM(3,10) is 408, attained on 179 orbits. This raises the best known lower bound on the covering radius of RM(2,10) from 400 to 408. A complementary heuristic search shows that the relative covering radius of RM(6,10) in RM(7,10) is at most 32, improving the previous bound of 50.

Problem

• Reed-Muller code
• weight distribution
• covering radius
• Boolean cubic forms
• nonlinearity

Innovation

• Reed-Muller codes
• weight distribution
• Boolean cubic forms
• covering radius
• nonlinearity