\([7,3,4]\) simplex code 

Second-smallest member of the simplex code family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(2,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((8,9)\) simplex spherical code under the antipodal mapping.

Its generator matrix is \begin{align} \left(\begin{array}{ccccccc} 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ \end{array}\right)~. \tag*{(1)}\end{align} The automorphism group of the code is \(GL_{3}(\mathbb{F}_{2})\), the second-smallest simple group.