The generator matrix

 1  0  0  1
 0  1  0  1
 0  0  1 a^2
 0  0  0  X

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

Homogenous weight enumerator: w(x)=1x^0+12x^4+288x^6+54x^8+1536x^9+1728x^10+5484x^12+4608x^13+2592x^14+81x^16

The gray image is a linear code over GF(4) with n=16, k=7 and d=4.
This code was found by Heurico 1.16 in 0.015 seconds.