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

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.