The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  X  X  X  X  X  X  0  0  0  0  0  0  0  1  1  X  1  1  1  1  1  1
 0  X  0  0  0  X  X  X  0  0  0  X  0  X  X  X  0  0  X  X  0  X  X  X  X  X  X  0  0  0  0  0  0  0  X  0  X  X  X
 0  0  X  0  X  X  X  0  0  0  X  X  X  X  0  0  0  X  X  0  X  X  0  0  0  X  X  X  X  0  0  0  0  X  X  X  X  0  0
 0  0  0  X  X  0  X  X  0  X  X  0  0  X  X  0  X  X  0  0  0  X  X  0  X  X  0  0  X  X  0  X  0  X  0  0  X  X  0

generates a code of length 39 over Z2[X]/(X^2) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+30x^40+1x^48

The gray image is a linear code over GF(2) with n=78, k=5 and d=40.
As d=40 is an upper bound for linear (78,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.00931 seconds.