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  X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  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^3  1  1  X  1  1  1  1
 0  X  0  X X^3 X^3 X^3+X X^3+X X^2 X^2+X X^2 X^2+X X^3+X^2 X^3+X^2+X X^3+X^2 X^3+X^2+X  0  X X^3+X^2 X^3+X^2+X  0  X X^3+X^2 X^3+X^2+X X^2  0 X^3+X X^3+X^2+X X^2 X^3+X^2+X  0 X^3+X X^2  0 X^3 X^3+X^2+X  X X^3+X^2+X X^3+X X^3+X^2 X^3+X^2  X X^3+X^2+X X^3 X^3 X^3 X^3 X^3+X X^2+X  X X^3+X^2+X X^2 X^2 X^2 X^3+X^2+X X^3+X  0 X^3 X^3+X^2+X X^2+X  X X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X X^3  0  0 X^2+X X^3+X X^3 X^3 X^2+X X^3+X^2+X  X X^2 X^3+X X^2+X X^3 X^3 X^3+X X^3+X^2 X^2+X  X X^2  X X^3+X^2+X
 0  0  X  X X^2 X^3+X^2+X X^2+X X^3+X^2 X^2 X^2+X  X  0  0  X X^3+X^2+X X^3+X^2  0 X^3+X  X X^3 X^2 X^3+X^2+X X^3+X^2+X X^2 X^2 X^2+X X^2 X^2+X  0  X X^3+X X^3 X^3+X^2+X X^3 X^2+X X^2 X^2+X X^3+X  0 X^3 X^2+X X^3+X  0 X^3+X^2  X  0 X^3+X^2+X  X X^3 X^3+X^2+X X^3+X^2 X^2+X X^3+X X^3+X^2 X^3+X^2+X X^2  X X^3+X^2 X^2+X  X X^3 X^2  0 X^3+X X^2+X X^3+X^2 X^3+X X^3 X^3+X^2 X^3+X^2 X^3+X  X  0 X^3+X^2 X^3 X^2+X X^3 X^3+X X^3+X X^2+X X^3 X^2+X X^3+X  0 X^3+X^2 X^3+X^2+X X^2  0
 0  0  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0  0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0  0  0  0  0  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0  0  0  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0  0  0  0 X^3 X^3 X^3  0 X^3 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+89x^84+148x^85+232x^86+504x^87+234x^88+508x^89+128x^90+48x^91+47x^92+52x^93+20x^94+8x^95+12x^96+12x^97+4x^98+1x^168

The gray image is a linear code over GF(2) with n=704, k=11 and d=336.
This code was found by Heurico 1.16 in 0.875 seconds.