The generator matrix

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

generates a code of length 7 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+18x^4+96x^5+109x^6+576x^7+109x^8+96x^9+18x^10+1x^14

The gray image is a code over GF(2) with n=56, k=10 and d=16.
This code was found by Heurico 1.16 in 3.81e-009 seconds.