The generator matrix

 1  1  X  1  1  1  1
 0 X^2+2 X^2+2  0  0  2  0
 0  0  2  0  2  2  0
 0  0  0  2  2  0  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+8x^4+40x^6+160x^7+37x^8+8x^10+2x^12

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 1.05e-007 seconds.