The generator matrix

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

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+594x^78+1862x^79+2822x^80+4582x^81+5892x^82+6654x^83+7238x^84+7532x^85+6798x^86+6708x^87+5091x^88+4014x^89+2657x^90+1690x^91+808x^92+264x^93+163x^94+86x^95+28x^96+4x^97+31x^98+8x^99+4x^100+4x^101+1x^102

The gray image is a linear code over GF(2) with n=680, k=16 and d=312.
This code was found by Heurico 1.16 in 46.3 seconds.