The generator matrix

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

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+578x^4+2480x^5+13501x^6+32416x^7+13501x^8+2480x^9+578x^10+1x^14

The gray image is a linear code over GF(2) with n=56, k=16 and d=16.
This code was found by Heurico 1.16 in 5.78 seconds.