On some minimal linear codes invariant under the Chevalley group \(\textrm{F}_4(2)\)

Minimal linear codes play a central role in applications such as secret sharing and secure communication, and their construction often reveals deep connections between coding theory and other areas of mathematics. Using a representation theoretic approach we construct six new minimal binary linear codes. Our approach exploits the action of the adjoint Chevalley group \(F_4(2)\) on its natural 26-dimensional module, yielding codes of very large length, small dimension, and large minimum distance. The resulting codes are projective, divisible, and invariant under \(F_4(2)\) acting transitively on coordinates. We provide a geometric description of minimum weight codewords and show that their stabilizers are maximal or second maximal subgroups of \(F_4(2)\) . Furthermore, we demonstrate that these codes provide examples of both narrow and wide minimal codes. Our work links the structure of an exceptional finite group to the construction of highly symmetric minimal codes, thereby enriching the interplay between group theory and coding theory.