The generator matrix

 1  0  0  1  1  1  1
 0  1  0  1 X^2+X X^2  0
 0  0  1  1 X+1 X^2+X+1  0
 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+78x^4+318x^5+1552x^6+4298x^7+1553x^8+306x^9+80x^10+6x^11

The gray image is a linear code over GF(2) with n=56, k=13 and d=16.
This code was found by Heurico 1.16 in 0.031 seconds.