The generator matrix

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

generates a code of length 88 over Z2[X]/(X^4) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+634x^84+104x^85+940x^86+176x^87+689x^88+128x^89+700x^90+80x^91+450x^92+24x^93+108x^94+27x^96+12x^98+20x^100+2x^112+1x^120

The gray image is a linear code over GF(2) with n=704, k=12 and d=336.
This code was found by Heurico 1.16 in 41.5 seconds.