Minimal binary linear codes are a special class of binary codes with important applications in secret sharing and secure two-party computation. These codes are characterized by the property that none ...of the nonzero codewords is covered by any other codeword. Denoting by
w
min
and
w
max
the minimum and maximum weights of the codewords, respectively, such codes are relatively easy to design when the ratio
w
min
/
w
max
is larger than 1/2 (known as the Ashikhmin-Barg bound). On the other hand, a few known classes of minimal codes violate this bound, hence having the property
w
min
/
w
max
≤
1
/
2
. In this article, we provide several
explicit
classes of minimal binary linear codes that violate the Ashikhmin-Barg bound while achieving a great variety of the ratio
w
min
/
w
max
. Our first generic method employs suitable characteristic functions with relatively low weights within the range
n
+ 1,2
n
− 2
. The second approach specifies characteristic functions with weights in 2
n
− 2
+ 1,2
n
− 2
+ 2
n
− 3
− 1, whose supports contain a skewed (removing one element) affine subspace of dimension
n
− 2. Finally, we also characterize an infinite family of minimal codes based on the class of so-called root Boolean functions of weight 2
n
− 1
− (
n
− 1), useful in specific hardware testing applications. Consequently, many infinite classes of minimal codes crossing the Ashikhmin-Barg bound are derived from an ample range of characteristic functions. In certain cases, we completely specify the weight distributions of the resulting codes.
