The generator matrix

 1  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  X
 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  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 a*X  0

generates a code of length 21 over F4[X]/(X^2) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+69x^56+54x^60+768x^63+66x^64+30x^68+15x^72+6x^76+9x^80+6x^84

The gray image is a linear code over GF(4) with n=84, k=5 and d=56.
This code was found by Heurico 1.16 in 0.016 seconds.