The generator matrix

 1  1  1  1  1  1  1  1  1  1  X  1  1  X  X  X X^2  X  1  X  1 X^2 X^2 X^3  0  1
 0 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0

generates a code of length 26 over Z2[X]/(X^4) who´s minimum homogenous weight is 26.

Homogenous weight enumerator: w(x)=1x^0+16x^26+10x^27+1x^28+2x^29+2x^30

The gray image is a linear code over GF(2) with n=208, k=5 and d=104.
As d=106 is an upper bound for linear (208,5,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 5.
This code was found by Heurico 1.16 in 0 seconds.