The generator matrix

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

generates a code of length 44 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 124.

Homogenous weight enumerator: w(x)=1x^0+171x^124+636x^125+243x^128+816x^129+222x^132+600x^133+225x^136+720x^137+129x^140+300x^141+18x^144+9x^152+6x^156

The gray image is a linear code over GF(4) with n=176, k=6 and d=124.
This code was found by Heurico 1.16 in 0.177 seconds.