The generator matrix

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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+535x^78+1632x^79+3765x^80+5974x^81+7979x^82+10390x^83+12914x^84+14498x^85+15683x^86+14954x^87+12980x^88+10988x^89+7753x^90+4960x^91+3174x^92+1484x^93+764x^94+306x^95+200x^96+72x^97+32x^98+14x^99+4x^100+6x^101+6x^102+2x^104+2x^105

The gray image is a linear code over GF(2) with n=688, k=17 and d=312.
This code was found by Heurico 1.16 in 206 seconds.