A Construction of Linear Codes Over ${\mathbb {F}}_{2^t}$ From Boolean Functions

Abstract

In this paper, we present a construction of linear codes over ${\mathbb {F}}_{2^{t}}$ from Boolean functions, which is a generalization of Ding’s method. Based on this construction, we give two classes of linear codes $\tilde { {\mathcal {C}}}_{f}$ and ${\mathcal {C}}_{f}$ over ${\mathbb {F}}_{2^{t}}$ from a Boolean function $f: {\mathbb {F}}_{q}\rightarrow {\mathbb {F}} _{2}$ , where $q=2^{n}$ and ${\mathbb {F}}_{2^{t}}$ is some subfield of ${\mathbb {F}}_{q}$ . The complete weight enumerator of $\tilde { {\mathcal {C}}}_{f}$ can be easily determined from the Walsh spectrum of $f$ , while the weight distribution of the code ${\mathcal {C}}_{f}$ can also be easily settled. Particularly, the number of nonzero weights of $\tilde { {\mathcal {C}}}_{f}$ and ${\mathcal {C}}_{f}$ 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 ${\mathbb {F}}_{2^{t}}$ with two or three weights by using bent, semibent, monomial and quadratic Boolean function $f$ .