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

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+124x^64+242x^65+231x^66+306x^67+366x^68+288x^69+163x^70+128x^71+95x^72+38x^73+37x^74+14x^75+5x^76+8x^77+1x^78+1x^116

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 0.36 seconds.