# On the weight distribution of second order Reed-Muller codes and their relatives

## TL;DR

This paper accurately determines the weight distribution of second order q-ary Reed-Muller codes and their variants, correcting previous errors and extending the results to homogeneous and projective versions.

## Contribution

It provides a precise computation of weight distributions for second order q-ary Reed-Muller codes and their related codes, correcting past inaccuracies.

## Key findings

- Corrected weight distribution formulas for second order q-ary Reed-Muller codes
- Extended weight distribution results to homogeneous Reed-Muller codes
- Extended weight distribution results to projective Reed-Muller codes

## Abstract

The weight distribution of second order $q$-ary Reed-Muller codes have been determined by Sloane and Berlekamp (IEEE Trans. Inform. Theory, vol. IT-16, 1970) for $q=2$ and by McEliece (JPL Space Programs Summary, vol. 3, 1969) for general prime power $q$. Unfortunately, there were some mistakes in the computation of the latter one. This paper aims to provide a precise account for the weight distribution of second order $q$-ary Reed-Muller codes. In addition, the weight distributions of second order $q$-ary homogeneous Reed-Muller codes and second order $q$-ary projective Reed-Muller codes are also determined.