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

TL;DR

This paper introduces a new method to construct linear codes over finite fields using Boolean functions, generalizing previous approaches and enabling the creation of codes with specific weight properties based on Walsh spectra.

Contribution

It generalizes Ding's construction of linear codes from Boolean functions to codes over subfields of _{2^t}, providing explicit formulas for weight enumerators and new classes of codes.

Findings

Complete weight enumerator of _{ ilde{C}_f} derived from Walsh spectrum.

Weight distribution of _{C_f} explicitly determined.

Constructed codes include two- and three-weight codes using bent, semibent, monomial, and quadratic 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 $f$.