The generator matrix

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

generates a code of length 7 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 12.

Homogenous weight enumerator: w(x)=1x^0+30x^12+153x^16+768x^18+396x^20+2304x^22+414x^24+30x^28

The gray image is a linear code over GF(4) with n=28, k=6 and d=12.
This code was found by Heurico 1.16 in 0.00237 seconds.