The generator matrix

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

generates a code of length 13 over F4[X]/(X^2) who´s minimum homogenous weight is 31.

Homogenous weight enumerator: w(x)=1x^0+384x^31+366x^32+384x^33+1152x^35+975x^36+2304x^37+2688x^39+2106x^40+3456x^41+1920x^43+606x^44+39x^48+3x^52

The gray image is a linear code over GF(4) with n=52, k=7 and d=31.
This code was found by Heurico 1.16 in 30.4 seconds.