The generator matrix

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

generates a code of length 13 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 28.

Homogenous weight enumerator: w(x)=1x^0+165x^28+228x^29+204x^30+924x^31+861x^32+1476x^33+1740x^34+3132x^35+4011x^36+6540x^37+5124x^38+9396x^39+9252x^40+9228x^41+5220x^42+4980x^43+2055x^44+960x^45+30x^48+9x^52

The gray image is a linear code over GF(4) with n=52, k=8 and d=28.
This code was found by Heurico 1.16 in 1.76 seconds.