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

generates a code of length 69 over Z2[X]/(X^4) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+183x^68+66x^72+5x^76+1x^88

The gray image is a linear code over GF(2) with n=552, k=8 and d=272.
This code was found by Heurico 1.16 in 1.69 seconds.