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  1  1  1  1  1  1  1  1  1  1  X  1  1  X  1  1  1  1  X  X
 0 X^3+X^2  0  0  0 X^2 X^3+X^2 X^2  0  0  0  0 X^2 X^3+X^2 X^2 X^3+X^2  0  0 X^2 X^2 X^3 X^2 X^3 X^2  0 X^3+X^2 X^3  0 X^3 X^2 X^2 X^3+X^2 X^3 X^3+X^2 X^2 X^3+X^2 X^3  0 X^2 X^3 X^2  0 X^2 X^3+X^2 X^3 X^3  0 X^3 X^3+X^2  0 X^3+X^2 X^3 X^3+X^2  0 X^2 X^3  0 X^3 X^3+X^2 X^3 X^2 X^3+X^2 X^3+X^2 X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2
 0  0 X^3+X^2  0 X^2 X^2 X^3+X^2  0 X^2  0  0 X^3+X^2 X^2 X^3+X^2  0  0  0 X^2 X^3+X^2 X^3  0 X^2 X^3+X^2 X^3 X^3 X^3+X^2 X^2 X^3 X^3+X^2 X^2  0 X^3 X^3+X^2 X^2 X^3  0  0 X^2 X^2  0 X^3+X^2  0 X^3+X^2 X^3 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0 X^3 X^3 X^3 X^3 X^3 X^3 X^2 X^2 X^2  0  0  0 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2  0 X^3
 0  0  0 X^3+X^2 X^2  0 X^3+X^2 X^2 X^2  0 X^3+X^2  0  0 X^3+X^2 X^2  0 X^3 X^3+X^2  0 X^2 X^3+X^2 X^2  0 X^3 X^3  0 X^3+X^2 X^2 X^3 X^3+X^2 X^3+X^2  0 X^2  0 X^3 X^3+X^2  0 X^3 X^3+X^2 X^2 X^3 X^2 X^3+X^2  0 X^3 X^2 X^3 X^3+X^2 X^3+X^2  0 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3 X^3 X^2 X^3  0  0  0 X^2 X^2 X^3 X^3 X^3  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 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0  0  0  0  0  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3  0  0  0  0  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0 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+334x^64+1408x^68+288x^72+16x^80+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 84.5 seconds.