The generator matrix
1 1 1 1 1 1 1 X 1 X^2 X
0 X 0 0 0 0 X^2 X^2+X X X X^2
0 0 X 0 0 X^2 X^2+X X X^2 X^2+X 0
0 0 0 X 0 X^2+X X X^2 X X^2+X X
0 0 0 0 X X 0 X^2+X X^2 X^2+X X^2
generates a code of length 11 over Z2[X]/(X^3) who´s minimum homogenous weight is 8.
Homogenous weight enumerator: w(x)=1x^0+362x^8+480x^10+1096x^12+32x^14+77x^16
The gray image is a linear code over GF(2) with n=44, k=11 and d=16.
As d=17 is an upper bound for linear (44,11,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 11.
This code was found by Heurico 1.16 in 0.281 seconds.