The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+84x^28+165x^32+180x^36+3072x^39+240x^40+204x^44+138x^48+12x^52

The gray image is a linear code over GF(4) with n=52, k=6 and d=28.
This code was found by Heurico 1.16 in 0.0206 seconds.