In this paper, for \(n \ge 6\) , we present the generic construction of binary linear codes of length \(2^n-1\) with dimension \(n+3\) , and derive the necessary and sufficient condition for the constructed codes to be minimal. Using this generic construction, a new family of minimal binary linear codes violating the Ashikhmin-Barg condition will be constructed from a special class of Boolean functions. We also obtain the weight distribution of the constructed minimal binary linear codes. We will achieve minimal codes with the highest dimension, resulting in a better rate of transmission.
