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 F 2 t from Boolean functions, which is a generalization of Ding's method. Based on this construction, we give two classes of linear codesC f and C f over F 2 t from a Boolean function f : F q → F 2 , where q = 2 n and F 2 t is some subfield of F q . The complete weight enumerator ofC 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 ofC 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 . Linear codes, Walsh spectrum, Boolean function, bent and semibent functions, weight enumerator.

A construction of codes with linearity from two linear codes

“…• Interesting linear codes from 2-designs and Boolean functions can be found in [10,11,32]. We refer the reader to [17,20,23,31,34] for other linear codes with a few weights or optimal parameters.…”