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

generates a code of length 82 over Z2[X]/(X^4) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+562x^78+288x^79+678x^80+304x^81+600x^82+320x^83+617x^84+192x^85+364x^86+32x^87+72x^88+16x^89+12x^90+21x^92+10x^94+4x^98+1x^104+2x^108

The gray image is a linear code over GF(2) with n=656, k=12 and d=312.
This code was found by Heurico 1.16 in 98.9 seconds.