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

generates a code of length 64 over Z2[X]/(X^3) who´s minimum homogenous weight is 54.

Homogenous weight enumerator: w(x)=1x^0+61x^54+130x^55+165x^56+274x^57+280x^58+306x^59+501x^60+612x^61+723x^62+728x^63+770x^64+766x^65+644x^66+594x^67+419x^68+354x^69+277x^70+168x^71+122x^72+88x^73+78x^74+52x^75+32x^76+18x^77+14x^78+6x^79+6x^80+2x^82+1x^94

The gray image is a linear code over GF(2) with n=256, k=13 and d=108.
This code was found by Heurico 1.16 in 5.28 seconds.