The generator matrix

 1  1  1  1  X  X  X
 0 X^3+X^2  0 X^3+X^2 X^3+X^2 X^3+X^2  0
 0  0 X^3  0  0  0 X^3
 0  0  0 X^3  0  0 X^3
 0  0  0  0 X^3  0 X^3
 0  0  0  0  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+45x^4+32x^5+72x^6+704x^7+107x^8+32x^9+24x^10+7x^12

The gray image is a linear code over GF(2) with n=56, k=10 and d=16.
This code was found by Heurico 1.16 in -3.24e-008 seconds.