The generator matrix

 1  1  1  1 2X  X  1
 0  X  0  X  X X+2  0
 0  0  X  X  2  0 2X
 0  0  0 2X  0 2X 2X

generates a code of length 7 over Z4[X]/(X^2+2) 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 code over GF(2) with n=56, k=11 and d=16.
This code was found by Heurico 1.16 in 3.62e-008 seconds.