The generator matrix

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

generates a code of length 22 over Z2[X]/(X^4) who´s minimum homogenous weight is 22.

Homogenous weight enumerator: w(x)=1x^0+24x^22+5x^24+2x^28

The gray image is a linear code over GF(2) with n=176, k=5 and d=88.
As d=90 is an upper bound for linear (176,5,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 5.
This code was found by Heurico 1.16 in 3.62e-008 seconds.