The generator matrix

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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+136x^79+829x^80+1618x^81+2159x^82+3152x^83+3449x^84+3656x^85+3799x^86+3544x^87+3068x^88+2744x^89+1764x^90+1260x^91+721x^92+396x^93+206x^94+96x^95+88x^96+30x^97+39x^98+4x^99+4x^100+4x^101+1x^110

The gray image is a linear code over GF(2) with n=688, k=15 and d=316.
This code was found by Heurico 1.16 in 18.7 seconds.