The generator matrix

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

generates a code of length 21 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+51x^44+243x^48+402x^52+192x^54+480x^56+1728x^58+513x^60+5184x^62+759x^64+5184x^66+801x^68+540x^72+228x^76+57x^80+21x^84

The gray image is a linear code over GF(4) with n=84, k=7 and d=44.
This code was found by Heurico 1.16 in 0.582 seconds.