A construction of binary linear codes from Boolean functions

Abstract

Boolean functions have important applications in cryptography and coding theory. Two famous classes of binary codes derived from Boolean functions are the Reed-Muller codes and Kerdock codes. In the past two decades, a lot of progress on the study of applications of Boolean functions in coding theory has been made. Two generic constructions of binary linear codes with Boolean functions have been well investigated in the literature. The objective of this paper is twofold. The first is to provide a survey on recent results, and the other is to propose open problems on one of the two generic constructions of binary linear codes with Boolean functions. These open problems are expected to stimulate further research on binary linear codes from Boolean functions.Its length is q − 1, and its dimension is at most 2m and is equal to 2m in many cases. The dual of C * ( f ) has dimension at least q − 1 − 2m. This is a generic construction of linear codes, which has a long history and its importance is supported by Delsarte's Theorem [24]. It gives a coding-theory characterisation of APN monomials, almost bent functions, and semibent functions (see, for examples, [13], [8] and [44]) when q = 2. We will not deal with this construction in this paper. The second generic construction of linear codes from functionsIn this section, we present the second generic construction of linear codes over GF(p) with any subset D of GF(p m ), and introduce basic results about the linear codes. In Section 5, we will consider specific families of binary linear codes from Boolean functions obtained with this generic construction.3

Bent functions linear on elements of some classical spreads and presemifields spreads

“…A bent function is a Boolean function with an even number of variables which achieves the maximum possible nonlinearity. Bent functions have attracted a lot of research for four decades because of their relation to coding theory (in particular, as explained by Ding in the two nice papers [11,12], bent functions give rise automatically to linear codes), sequences, applications in cryptography and other domains such as combinatorics and design theory.…”

Several Classes of Minimal Linear Codes With Few Weights From Weakly Regular Plateaued Functions

“…f of the code C D f of Theorem 2 is the threeweight linear code with parameters (p n−1 − 1)/(p − 1), n, p n−2 − p (n+s−3)/2 p whose weight distribution is listed in Table 4. (1,8,6,12), which is verified by MAGMA in [2]. This code is optimal owing to the Singleton bound.…”

“…for all β ∈ F 3 3 , where ǫ = −1 and η 0 (−1) = −1. Then, C D f is the three-weight linear code with parameters [8, 3, 4] 3 , weight enumerator 1 + 8y 6 + 6y 8 + 12y 4 and weight distribution (1,8,6,12), which is verified by MAGMA in [2].…”

“…. , Tr n (βd m )) : β ∈ Supp( χ f )}, (6) which is the subcode of C D f defined by (5). Hence, the following codes in Corollaries 3, 4, 5 and 6 are the subcodes of the codes of Theorems 1, 2 and Corollaries 1, 2, respectively.…”

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.…”