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

generates a code of length 71 over Z2[X]/(X^4) who´s minimum homogenous weight is 65.

Homogenous weight enumerator: w(x)=1x^0+48x^65+45x^66+70x^67+52x^68+228x^69+417x^70+362x^71+425x^72+208x^73+39x^74+62x^75+28x^76+20x^77+11x^78+14x^79+4x^80+8x^81+4x^83+1x^88+1x^128

The gray image is a linear code over GF(2) with n=568, k=11 and d=260.
This code was found by Heurico 1.16 in 1.06 seconds.