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

generates a code of length 80 over Z2[X]/(X^2) who´s minimum homogenous weight is 81.

Homogenous weight enumerator: w(x)=1x^0+16x^81+6x^82+6x^84+1x^86+1x^88+1x^94

The gray image is a linear code over GF(2) with n=160, k=5 and d=81.
As d=81 is an upper bound for linear (160,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.116 seconds.