The generator matrix

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

generates a code of length 5 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+120x^6+30x^8+960x^9+1080x^10+6780x^12+5760x^13+3240x^14+15360x^15+20265x^16+8640x^17+3240x^18+60x^20

The gray image is a linear code over GF(4) with n=20, k=8 and d=6.
This code was found by Heurico 1.16 in 0.0894 seconds.