The generator matrix

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

generates a code of length 20 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+180x^44+315x^48+426x^52+639x^56+13068x^60+798x^64+570x^68+321x^72+60x^76+6x^80

The gray image is a linear code over GF(4) with n=80, k=7 and d=44.
This code was found by Heurico 1.16 in 0.576 seconds.