  Recently, minimal linear codes have been extensively studied due to their
applications in secret sharing schemes, secure two-party computations, and so
on. Constructing minimal linear codes violating the Ashikhmin-Barg condition
and then determining their weight distributions have been interesting in coding
theory and cryptography. In this paper, a generic construction for binary
linear codes with dimension $m+2$ is presented, then a necessary and sufficient
condition for this binary linear code to be minimal is derived. Based on this
condition and exponential sums, a new class of minimal binary linear codes
violating the Ashikhmin-Barg condition is obtained, and then their weight
enumerators are determined.
