The generator matrix
1 1 1 1 X X
0 X 0 X^2+X X X^2+X
0 0 X^2 0 0 X^2
0 0 0 X^2 0 X^2
0 0 0 0 X^2 X^2
generates a code of length 6 over Z2[X]/(X^3) who´s minimum homogenous weight is 4.
Homogenous weight enumerator: w(x)=1x^0+55x^4+144x^6+55x^8+1x^12
The gray image is a linear code over GF(2) with n=24, k=8 and d=8.
As d=8 is an upper bound for linear (24,8,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 8.
This code was found by Heurico 1.16 in 0.000207 seconds.