The generator matrix
1 1 1 1 1 1 1 1 X 0 X
0 X 0 X^2+X X^2 X^2+X X^2 X X^2+X X 0
0 0 X^2 X^2 X^2 0 0 X^2 0 X^2 0
generates a code of length 11 over Z2[X]/(X^3) who´s minimum homogenous weight is 10.
Homogenous weight enumerator: w(x)=1x^0+30x^10+30x^12+2x^14+1x^16
The gray image is a linear code over GF(2) with n=44, k=6 and d=20.
As d=21 is an upper bound for linear (44,6,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 6.
This code was found by Heurico 1.16 in 0.00045 seconds.