The generator matrix

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

generates a code of length 85 over Z2[X]/(X^4) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+214x^78+788x^79+1499x^80+2330x^81+2798x^82+3458x^83+3906x^84+3822x^85+3671x^86+3118x^87+2287x^88+2000x^89+1384x^90+738x^91+369x^92+194x^93+80x^94+30x^95+21x^96+18x^97+11x^98+12x^99+5x^100+4x^101+9x^102+1x^106

The gray image is a linear code over GF(2) with n=680, k=15 and d=312.
This code was found by Heurico 1.16 in 17.7 seconds.