The generator matrix
1 0 0 0 1 1 1 1 X^2 1 1 1
0 1 0 0 0 X+1 X^2+1 1 1 1 X^2+1 X^2+X+1
0 0 1 0 1 X^2+1 1 X^2 1 X+1 X+1 1
0 0 0 1 1 X^2+X+1 0 1 X^2+1 X+1 X+1 X+1
0 0 0 0 X X X^2+X X^2+X X X X^2 X^2
generates a code of length 12 over Z2[X]/(X^3) who´s minimum homogenous weight is 8.
Homogenous weight enumerator: w(x)=1x^0+622x^8+256x^9+2616x^10+1792x^11+5764x^12+1792x^13+2736x^14+256x^15+521x^16+24x^18+4x^20
The gray image is a linear code over GF(2) with n=48, k=14 and d=16.
As d=17 is an upper bound for linear (48,14,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 14.
This code was found by Heurico 1.13 in 427 seconds.