The generator matrix
1 0 1 1 1 X^2 X
0 1 1 0 X+1 1 X^2
0 0 X 0 0 X^2 X
0 0 0 X 0 X X^2
0 0 0 0 X^2 X^2 0
generates a code of length 7 over Z2[X]/(X^3) who´s minimum homogenous weight is 4.
Homogenous weight enumerator: w(x)=1x^0+109x^4+96x^5+648x^6+320x^7+683x^8+96x^9+88x^10+7x^12
The gray image is a linear code over GF(2) with n=28, k=11 and d=8.
As d=8 is an upper bound for linear (28,11,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 11.
This code was found by Heurico 1.16 in 0.00287 seconds.