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

generates a code of length 88 over Z2[X]/(X^4) who´s minimum homogenous weight is 83.

Homogenous weight enumerator: w(x)=1x^0+80x^83+62x^84+138x^85+166x^86+364x^87+449x^88+358x^89+148x^90+128x^91+62x^92+78x^93+6x^94+4x^95+1x^96+2x^97+1x^168

The gray image is a linear code over GF(2) with n=704, k=11 and d=332.
This code was found by Heurico 1.16 in 0.938 seconds.