Minimal linear codes have significant applications in secret sharing schemes, secure multi-party computation, and cryptography. In this paper, we propose a generic construction of a new family of minimal binary linear codes with dimension n+4 from a special class of Boolean functions. By leveraging the geometric properties of partial spreads in finite fields, we determine the explicit weight distribution and weight enumerator of the constructed codes. Furthermore, we derive a necessary and sufficient condition for these codes to be minimal, and establish that the proposed family yields minimal codes that structurally violate the well-known Ashikhmin-Barg condition, making them highly desirable for advanced communication systems.

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