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

generates a code of length 43 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 116.

Homogenous weight enumerator: w(x)=1x^0+117x^116+249x^120+612x^124+1428x^128+1395x^132+72x^136+93x^140+45x^144+27x^148+33x^152+9x^156+12x^160+3x^164

The gray image is a linear code over GF(4) with n=172, k=6 and d=116.
This code was found by Heurico 1.16 in 0.123 seconds.