The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  X  X  X  1  1  1  X  X X^2 X^2 X^2  X  X  X  1  1 X^2
 0 X^3  0 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0
 0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3  0 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+56x^28+7x^32

The gray image is a linear code over GF(2) with n=224, k=6 and d=112.
As d=112 is an upper bound for linear (224,6,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 6.
This code was found by Heurico 1.16 in 0.016 seconds.