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 a*X  X  0  X a^2*X a^2*X a*X a^2*X  0  X  0
 0  0  X  0  0  0  X a^2*X a^2*X  0  X a^2*X  X a^2*X  X  X a^2*X  X a^2*X a*X
 0  0  0  X  0  0 a^2*X  X  0 a*X a*X a^2*X  0 a*X a^2*X  X  X a*X  0 a^2*X
 0  0  0  0  X  0  X a^2*X a*X a^2*X  0 a*X a^2*X  0  X  0  X  X  0 a*X
 0  0  0  0  0  X a^2*X  X a^2*X a^2*X  0  0 a^2*X a^2*X a*X a*X  X a^2*X a*X  X

generates a code of length 20 over F4[X]/(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 0.571 seconds.