A Construction of Linear Codes over $\f_{2^t}$ from Boolean Functions

Abstract: In this paper, we present a construction of linear codes over $\f_{2^t}$ from Boolean functions, which is a generalization of Ding's method \cite[Theorem 9]{Ding15}. Based on this construction, we give two classes of linear codes $\tilde{\C}{f}$ and $\C_f$ (see Theorem \ref{thm-maincode1} and Theorem \ref{thm-maincodenew}) over $\f{2^t}$ from a Boolean function $f:\f_{q}\rightarrow \f_2$, where $q=2^n$ and $\f_{2^t}$ is some subfield of $\f_{q}$. The complete weight enumerator of $\tilde{\C}{f}$ can be easily determined from the Walsh spectrum of $f$, while the weight distribution of the code $\C_f$ can also be easily settled. Particularly, the number of nonzero weights of $\tilde{\C}{f}$ and $\C_f$ is the same as the number of distinct Walsh values of $f$. As applications of this construction, we show several series of linear codes over $\f_{2^t}$ with two or three weights by using bent, semibent, monomial and quadratic Boolean function $f$.