The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 0  X  0  0  X  0 a*X a^2*X a*X  X  X a^2*X a*X a*X a^2*X a^2*X
 0  0  X  0 a^2*X  X  0 a^2*X a*X a^2*X  0 a^2*X  X a^2*X  X  X
 0  0  0  X  X  X  X a^2*X a^2*X a^2*X a*X  0 a^2*X  X  0 a^2*X

generates a code of length 16 over F4[X]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+144x^44+828x^48+48x^60+3x^64

The gray image is a linear code over GF(4) with n=64, k=5 and d=44.
This code was found by Heurico 1.16 in 0.063 seconds.