The generator matrix

 1  0  0  1  1
 0  1  0  1  0
 0  0  1 a+1  0
 0  0  0  X  0
 0  0  0  0  X

generates a code of length 5 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+15x^4+144x^6+90x^8+960x^9+1296x^10+4878x^12+5760x^13+3888x^14+21504x^15+14229x^16+8640x^17+3888x^18+243x^20

The gray image is a linear code over GF(4) with n=20, k=8 and d=4.
This code was found by Heurico 1.16 in 0.17 seconds.