The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 X 1 1 X^2
0 X 0 X^2+X 0 X^2+X X^2 X^2+X 0 X^2+X 0 X^2+X 0 X^2 X^2 X X X^2
0 0 X^2 0 0 0 X^2 0 0 X^2 X^2 X^2 0 X^2 0 X^2 X^2 0
0 0 0 X^2 0 0 0 X^2 X^2 X^2 X^2 X^2 X^2 X^2 X^2 X^2 0 0
0 0 0 0 X^2 X^2 X^2 X^2 X^2 0 0 X^2 0 X^2 0 X^2 0 0
generates a code of length 18 over Z2[X]/(X^3) who´s minimum homogenous weight is 16.
Homogenous weight enumerator: w(x)=1x^0+58x^16+64x^17+32x^18+64x^19+24x^20+12x^24+1x^32
The gray image is a linear code over GF(2) with n=72, k=8 and d=32.
As d=32 is an upper bound for linear (72,8,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 8.
This code was found by Heurico 1.16 in 0.00284 seconds.