The generator matrix
1 0 0 0 0 1 1
0 1 0 0 0 1 0
0 0 1 0 0 1 0
0 0 0 1 0 1 0
0 0 0 0 1 1 0
0 0 0 0 0 X X^2
generates a code of length 7 over Z2[X]/(X^3) who´s minimum homogenous weight is 2.
Homogenous weight enumerator: w(x)=1x^0+127x^2+600x^3+3261x^4+11040x^5+28355x^6+44304x^7+28355x^8+11040x^9+3261x^10+600x^11+127x^12+1x^14
The gray image is a linear code over GF(2) with n=28, k=17 and d=4.
This code was found by Heurico 1.13 in 0.016 seconds.