The generator matrix

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

generates a code of length 6 over Z2[X]/(X^3) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+12x^6+3x^8

The gray image is a linear code over GF(2) with n=24, k=4 and d=12.
As d=12 is an upper bound for linear (24,4,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 4.
This code was found by Heurico 1.16 in 0.000109 seconds.