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

generates a code of length 89 over Z2[X]/(X^4) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+17x^80+32x^82+37x^84+16x^85+33x^86+64x^87+287x^88+1120x^89+280x^90+64x^91+16x^92+16x^93+13x^94+8x^96+14x^98+11x^100+8x^102+7x^104+2x^106+1x^110+1x^166

The gray image is a linear code over GF(2) with n=712, k=11 and d=320.
This code was found by Heurico 1.16 in 0.719 seconds.