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

generates a code of length 44 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 112.

Homogenous weight enumerator: w(x)=1x^0+153x^112+375x^116+444x^120+1035x^124+3027x^128+5688x^132+4332x^136+438x^140+369x^144+291x^148+168x^152+39x^156+18x^160+6x^164

The gray image is a linear code over GF(4) with n=176, k=7 and d=112.
This code was found by Heurico 1.16 in 1.82 seconds.