The generator matrix

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

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+309x^128+480x^132+192x^136+36x^144+6x^160

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 0.078 seconds.