The generator matrix

 1  1  1  1
 0  X  0  0
 0  0  X  0
 0  0  0  X

generates a code of length 4 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+12x^4+54x^8+876x^12+81x^16

The gray image is a linear code over GF(4) with n=16, k=5 and d=4.
This code was found by Heurico 1.16 in -3.24e-008 seconds.