The generator matrix

 1  0  1  1  1  1  1  1  1  1  1  1  1  X  X  0
 0  1  1  a a^2*X+a^2  0 a^2*X+1 a^2*X+a^2  X  a a*X+1 a^2*X+a a^2  X  1  1
 0  0 a^2*X a*X  X  X  0 a^2*X a*X a^2*X a*X  X  0  X  0 a*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+369x^44+279x^48+372x^52+3x^60

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 -1.01e-007 seconds.