The generator matrix

 1  1  1  1 X^3  X  1
 0  X  0  X  X X^2+X  0
 0  0  X  X X^2  0 X^3
 0  0  0 X^3  0 X^3 X^3

generates a code of length 7 over Z2[X]/(X^4) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+30x^4+78x^5+512x^6+810x^7+513x^8+66x^9+32x^10+6x^11

The gray image is a linear code over GF(2) with n=56, k=11 and d=16.
This code was found by Heurico 1.16 in -1.01e-007 seconds.