The generator matrix

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

generates a code of length 13 over F4[X]/(X^2) who´s minimum homogenous weight is 32.

Homogenous weight enumerator: w(x)=1x^0+51x^32+78x^36+768x^39+48x^40+48x^44+12x^48+18x^52

The gray image is a linear code over GF(4) with n=52, k=5 and d=32.
This code was found by Heurico 1.16 in 0.016 seconds.