The generator matrix
1 0 0 0 0 1 1 1 X
0 1 0 0 0 1 0 X^2+X X^2+X
0 0 1 0 0 1 0 X+1 1
0 0 0 1 0 1 1 X+1 X^2
0 0 0 0 1 1 X+1 X 1
0 0 0 0 0 X X^2+X X X^2+X
generates a code of length 9 over Z2[X]/(X^3) who´s minimum homogenous weight is 4.
Homogenous weight enumerator: w(x)=1x^0+240x^4+904x^5+5016x^6+13032x^7+29602x^8+33552x^9+29456x^10+13072x^11+5096x^12+872x^13+216x^14+8x^15+5x^16
The gray image is a linear code over GF(2) with n=36, k=17 and d=8.
As d=9 is an upper bound for linear (36,17,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 17.
This code was found by Heurico 1.13 in 0.765 seconds.