The generator matrix
1 0 0 0 1 X
0 1 0 0 1 X^2
0 0 1 0 1 0
0 0 0 1 1 0
0 0 0 0 X X
0 0 0 0 0 X^2
generates a code of length 6 over Z2[X]/(X^3) who´s minimum homogenous weight is 2.
Homogenous weight enumerator: w(x)=1x^0+146x^2+640x^3+3375x^4+7040x^5+10364x^6+7040x^7+3375x^8+640x^9+146x^10+1x^12
The gray image is a linear code over GF(2) with n=24, k=15 and d=4.
As d=4 is an upper bound for linear (24,15,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 15.
This code was found by Heurico 1.16 in 0.215 seconds.