The generator matrix

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

generates a code of length 88 over Z2[X]/(X^4) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+206x^82+8x^83+352x^84+200x^85+478x^86+576x^87+617x^88+544x^89+459x^90+184x^91+186x^92+24x^93+147x^94+71x^96+13x^98+18x^100+7x^102+2x^104+2x^106+1x^152

The gray image is a linear code over GF(2) with n=704, k=12 and d=328.
This code was found by Heurico 1.16 in 30.4 seconds.