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

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 83.

Homogenous weight enumerator: w(x)=1x^0+184x^83+89x^84+242x^85+303x^86+474x^87+304x^88+208x^89+46x^90+140x^91+17x^92+18x^93+3x^94+2x^95+4x^96+12x^97+1x^160

The gray image is a linear code over GF(2) with n=696, k=11 and d=332.
This code was found by Heurico 1.16 in 58.7 seconds.