The generator matrix
1 0 0 0 1 1 1 0
0 1 0 0 0 1 X^2+1 1
0 0 1 0 1 1 0 1
0 0 0 1 1 X^2+X 1 1
0 0 0 0 X 0 X^2+X X^2
0 0 0 0 0 X^2 0 X^2
generates a code of length 8 over Z2[X]/(X^3) who´s minimum homogenous weight is 4.
Homogenous weight enumerator: w(x)=1x^0+268x^4+768x^5+4032x^6+5376x^7+11878x^8+5376x^9+4032x^10+768x^11+268x^12+1x^16
The gray image is a linear code over GF(2) with n=32, k=15 and d=8.
As d=8 is an upper bound for linear (32,15,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 15.
This code was found by Heurico 1.16 in 70.2 seconds.