The generator matrix

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

generates a code of length 16 over Z2[X]/(X^3) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+28x^16+2x^20+1x^24

The gray image is a linear code over GF(2) with n=64, k=5 and d=32.
As d=32 is an upper bound for linear (64,5,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 0.000795 seconds.