The generator matrix

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

generates a code of length 17 over Z2[X]/(X^4) who´s minimum homogenous weight is 17.

Homogenous weight enumerator: w(x)=1x^0+22x^17+2x^18+6x^19+1x^20

The gray image is a linear code over GF(2) with n=136, k=5 and d=68.
As d=69 is an upper bound for linear (136,5,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 5.
This code was found by Heurico 1.16 in -3.24e-008 seconds.