The generator matrix
1 1 1 1 X X 1
0 X 0 X^2+X X X^2+X 0
0 0 X^2 0 0 X^2 0
0 0 0 X^2 0 X^2 0
0 0 0 0 X^2 X^2 0
generates a code of length 7 over Z2[X]/(X^3) who´s minimum homogenous weight is 4.
Homogenous weight enumerator: w(x)=1x^0+18x^4+32x^5+13x^6+128x^7+13x^8+32x^9+18x^10+1x^14
The gray image is a linear code over GF(2) with n=28, k=8 and d=8.
This code was found by Heurico 1.16 in 0.00024 seconds.