The generator matrix

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

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+50x^28+12x^30+1x^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 -6.48e-008 seconds.