The generator matrix

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

generates a code of length 44 over F4[X]/(X^2) who´s minimum homogenous weight is 128.

Homogenous weight enumerator: w(x)=1x^0+189x^128+768x^132+63x^144+3x^176

The gray image is a linear code over GF(4) with n=176, k=5 and d=128.
This code was found by Heurico 1.16 in 6.41 seconds.