The generator matrix

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

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+230x^83+835x^84+854x^85+1364x^86+580x^87+1202x^88+656x^89+898x^90+346x^91+451x^92+210x^93+248x^94+132x^95+100x^96+40x^97+9x^98+24x^99+8x^100+2x^104+1x^112+1x^114

The gray image is a linear code over GF(2) with n=704, k=13 and d=332.
This code was found by Heurico 1.16 in 6.06 seconds.