The generator matrix

 1  1  1  1  X  X  X
 0 X^2  0 X^2  0 X^2  0
 0  0 X^2 X^2 X^2 X^2  0

generates a code of length 7 over Z2[X]/(X^3) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+6x^6+16x^7+7x^8+2x^10

The gray image is a linear code over GF(2) with n=28, k=5 and d=12.
This code was found by Heurico 1.16 in 0.00013 seconds.