The generator matrix

 1  0  1  1  1  1  1  1  1  1  1  1  1
 0  1  1  a  0 a^2*X+1  X a^2*X+1  a X+a  X  1 X+a
 0  0 a^2*X a*X  X  0 a*X  X a^2*X  0 a^2*X a*X  X

generates a code of length 13 over F4[X]/(X^2) who´s minimum homogenous weight is 35.

Homogenous weight enumerator: w(x)=1x^0+216x^35+69x^36+336x^39+180x^40+216x^43+3x^48+3x^52

The gray image is a linear code over GF(4) with n=52, k=5 and d=35.
This code was found by Heurico 1.16 in 0.078 seconds.