The generator matrix
1 0 1 1 1 X^2+X 1 1 1 1
0 1 X+1 X^2+X X^2+1 1 0 X^2+X X+1 X^2+1
0 0 X^2 0 X^2 0 0 X^2 X^2 0
0 0 0 X^2 X^2 X^2 X^2 X^2 X^2 0
generates a code of length 10 over Z2[X]/(X^3) who´s minimum homogenous weight is 8.
Homogenous weight enumerator: w(x)=1x^0+51x^8+32x^9+88x^10+32x^11+50x^12+2x^16
The gray image is a linear code over GF(2) with n=40, k=8 and d=16.
As d=16 is an upper bound for linear (40,8,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 8.
This code was found by Heurico 1.16 in 0.000626 seconds.