The generator matrix

 1  1  1  1  1  1  1
 0  2  0  0  0  2  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+10x^4+5x^6+224x^7+5x^8+10x^10+1x^14

The gray image is a code over GF(2) with n=56, k=8 and d=16.
This code was found by Heurico 1.16 in 0.016 seconds.