The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 X^3  1  1  1  1  1  1  1  1  X  1  1  1  X  1  1 X^3  1  1  1  0 X^3+X^2  1  1 X^2  1 X^3+X^2  1  X X^3+X^2  X  1 X^2  X  X  X  0  X  X
 0  X  0 X^3+X^2+X X^2 X^2+X X^3+X^2  X  0 X^2+X X^3 X^2+X X^3+X X^2 X^2  X  0 X^2+X X^2 X^3+X X^3 X^3+X X^3 X^3+X X^2+X  0 X^3  X X^3 X^2+X X^2+X  0 X^2 X^3+X X^3+X^2+X X^3+X^2 X^3+X  X X^3 X^3+X X^2 X^2  X  X X^3+X^2  0 X^3+X^2+X X^3  X X^2+X X^3+X^2+X X^3+X^2+X  X  X  X  X X^2+X  X  X X^3+X^2+X X^2+X  X  X  X  X X^2+X  X X^3+X X^2+X  X X^3+X^2+X X^2+X  X  X X^2 X^3+X^2+X
 0  0 X^3+X^2  0 X^2  0 X^3  0 X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0 X^2  0 X^3+X^2 X^3 X^3+X^2 X^2 X^3 X^2  0  0  0 X^2  0 X^3 X^2 X^2 X^2 X^3 X^2 X^3 X^2 X^3 X^2  0 X^3+X^2 X^3  0  0 X^3 X^3+X^2 X^2  0 X^3+X^2 X^3 X^2 X^3+X^2 X^2 X^2 X^3  0 X^2  0  0 X^3 X^2 X^3 X^3  0 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3 X^3 X^2 X^3  0  0 X^2 X^3  0
 0  0  0 X^3+X^2  0 X^3 X^3 X^2 X^2 X^2 X^2  0  0 X^2 X^3+X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^3  0 X^3+X^2 X^3+X^2 X^3  0 X^3+X^2  0 X^3+X^2  0 X^3 X^2 X^2 X^3 X^3+X^2 X^2  0 X^2 X^3  0  0 X^2 X^2  0  0 X^2 X^3 X^3+X^2 X^2 X^3+X^2  0 X^3 X^3+X^2 X^3 X^2 X^3+X^2 X^3+X^2 X^2  0 X^3 X^3  0 X^2 X^3 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3 X^3  0 X^2 X^3+X^2 X^3+X^2 X^2 X^2 X^2
 0  0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3  0  0 X^3  0  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3  0  0  0 X^3 X^3 X^3  0 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0

generates a code of length 76 over Z2[X]/(X^4) who´s minimum homogenous weight is 70.

Homogenous weight enumerator: w(x)=1x^0+108x^70+164x^71+381x^72+210x^73+543x^74+286x^75+799x^76+270x^77+499x^78+210x^79+364x^80+116x^81+89x^82+4x^83+16x^84+6x^85+8x^86+6x^87+6x^88+6x^89+2x^91+1x^102+1x^116

The gray image is a linear code over GF(2) with n=608, k=12 and d=280.
This code was found by Heurico 1.16 in 1.02 seconds.