The generator matrix

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

generates a code of length 29 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+78x^68+309x^72+417x^76+192x^78+468x^80+1728x^82+594x^84+5184x^86+591x^88+5184x^90+594x^92+552x^96+300x^100+156x^104+33x^108+3x^112

The gray image is a linear code over GF(4) with n=116, k=7 and d=68.
This code was found by Heurico 1.16 in 1.02 seconds.