The generator matrix

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

generates a code of length 78 over Z2[X]/(X^4) who´s minimum homogenous weight is 70.

Homogenous weight enumerator: w(x)=1x^0+247x^70+1546x^71+3081x^72+5644x^73+7018x^74+11332x^75+12620x^76+16328x^77+15416x^78+16748x^79+12677x^80+11512x^81+7057x^82+5122x^83+2358x^84+1306x^85+545x^86+278x^87+126x^88+56x^89+19x^90+14x^91+16x^92+2x^93+2x^94+1x^96

The gray image is a linear code over GF(2) with n=624, k=17 and d=280.
This code was found by Heurico 1.16 in 183 seconds.