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

generates a code of length 68 over Z2[X]/(X^4) who´s minimum homogenous weight is 64.

Homogenous weight enumerator: w(x)=1x^0+108x^64+300x^65+416x^66+304x^67+100x^68+172x^69+292x^70+160x^71+38x^72+52x^73+24x^74+16x^75+24x^76+20x^77+20x^78+1x^128

The gray image is a linear code over GF(2) with n=544, k=11 and d=256.
This code was found by Heurico 1.16 in 0.578 seconds.