The generator matrix

 1  0  1  1  1  1  1  1  1  0  0  1  1  1  1  1  1  1  1  1
 0  1  1  a a+1  0 (a+1)X+1  a (a+1)X+a+1  1  1  0 (a+1)X+a+1 (a+1)X+1 (a+1)X+a+1  a (a+1)X+1 (a+1)X+1  0 (a+1)X+a+1
 0  0 (a+1)X  0  0  0  0  X aX  X  X  X (a+1)X (a+1)X (a+1)X  X (a+1)X  0 (a+1)X  X
 0  0  0  X  0  0  X (a+1)X (a+1)X aX (a+1)X (a+1)X (a+1)X (a+1)X (a+1)X  X  0 (a+1)X (a+1)X aX
 0  0  0  0 (a+1)X  0 aX  0  X  X (a+1)X aX  X aX aX  0 (a+1)X (a+1)X  0  X
 0  0  0  0  0  X aX  X aX aX aX  0  0  0  X  X aX  X  X  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+168x^44+24x^45+402x^48+360x^49+384x^51+945x^52+2160x^53+3456x^55+3240x^56+6480x^57+10368x^59+6003x^60+9720x^61+10368x^63+4701x^64+5832x^65+567x^68+252x^72+93x^76+12x^80

The gray image is a linear code over GF(4) with n=80, k=8 and d=44.
This code was found by Heurico 1.16 in 4.42 seconds.