The generator matrix

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

generates a code of length 84 over Z2[X]/(X^4) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+478x^76+1564x^77+3612x^78+5870x^79+8027x^80+9906x^81+13651x^82+14748x^83+15626x^84+14700x^85+13685x^86+10766x^87+7740x^88+5000x^89+2989x^90+1336x^91+774x^92+314x^93+175x^94+48x^95+30x^96+2x^97+16x^98+10x^100+2x^101+2x^104

The gray image is a linear code over GF(2) with n=672, k=17 and d=304.
This code was found by Heurico 1.16 in 199 seconds.