The generator matrix

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

generates a code of length 64 over Z2[X]/(X^3) who´s minimum homogenous weight is 65.

Homogenous weight enumerator: w(x)=1x^0+16x^65+3x^66+8x^67+3x^68+1x^70

The gray image is a linear code over GF(2) with n=256, k=5 and d=130.
As d=131 is an upper bound for linear (256,5,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 0.106 seconds.