On $B$-Functions Encountered in Modular Codes

Assume that $f(x)$ is a function from $GF(2^m)$ to $GF(2)$. Function $f(x)$ is called a $B$-function if all its Fourier–Hadamard coefficients $\sum_u(-1)^{ux+f(x)}$ are equal to $\pm2^{m/2}$. $B$-functions play an important part in coding theory, particularly in the creation of Kerdock and Hadamard codes. The problem of classifying $B$-functions of degree 3 is considered.