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+1)X (a+1)X (a+1)X (a+1)X (a+1)X (a+1)X  0  0  X  X  0  X  0 aX  X  0 (a+1)X (a+1)X (a+1)X  X (a+1)X aX aX aX (a+1)X aX  0 (a+1)X  0 aX aX  0  X  X aX  X
 0  0  X  0 (a+1)X  X (a+1)X  0 (a+1)X (a+1)X  X  X (a+1)X  X  0  X (a+1)X  0  0 aX  X  X  0 (a+1)X  0 aX  X aX (a+1)X aX  X aX aX  0 aX  0 aX (a+1)X (a+1)X (a+1)X  X  X  0 (a+1)X
 0  0  0  X  X  X aX (a+1)X  0  X  0  X aX aX aX (a+1)X  0 aX (a+1)X (a+1)X aX  0  X  X aX  X (a+1)X aX (a+1)X  X (a+1)X aX (a+1)X aX  X  0  0 aX  X  0  0 aX  X  0

generates a code of length 44 over F4[X,sigma]/(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 6.5 seconds.