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

generates a code of length 20 over F4[X]/(X^2) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+36x^52+87x^56+810x^60+54x^64+12x^68+9x^72+6x^76+9x^80

The gray image is a linear code over GF(4) with n=80, k=5 and d=52.
This code was found by Heurico 1.16 in 6.87e-008 seconds.