The generator matrix

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

generates a code of length 22 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+264x^56+192x^60+1515x^64+1728x^68+264x^72+132x^80

The gray image is a linear code over GF(4) with n=88, k=6 and d=56.
This code was found by Heurico 1.16 in 49.8 seconds.