Distance regularity of Kerdock codes

A code is called distance regular, if for every two codewords $\mathbf x,\mathbf y$ and integers $i,j$ the number of codewords $\mathbf z$ such that $d(\mathbf x,\mathbf z)=i$ and $d(\mathbf y,\mathbf z)=j$, with $d$ the Hamming distance, does not depend on the choice of $\mathbf x,\mathbf y$ and depends only on $d(\mathbf x,\mathbf y)$ and $i,j$. Using some properties of the discrete Fourier transform we give a new combinatorial proof of the distance regularity of an arbitrary Kerdock code. We also calculate the parameters of the distance regularity of a Kerdock code.