The generator matrix

 1  0  0  1
 0  1  0 (a+1)X
 0  0  1  0
 0  0  0  1

generates a code of length 4 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 3.

Homogenous weight enumerator: w(x)=1x^0+48x^3+12x^4+864x^6+432x^7+54x^8+6912x^9+5184x^10+1296x^11+20844x^12+20736x^13+7776x^14+1296x^15+81x^16

The gray image is a linear code over GF(4) with n=16, k=8 and d=3.
This code was found by Heurico 1.16 in 0.0431 seconds.