The generator matrix

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

generates a code of length 28 over Z2[X]/(X^4) who´s minimum homogenous weight is 28.

Homogenous weight enumerator: w(x)=1x^0+16x^28+8x^29+5x^30+1x^32+1x^34

The gray image is a linear code over GF(2) with n=224, k=5 and d=112.
As d=114 is an upper bound for linear (224,5,2)-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.