The generator matrix

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

generates a code of length 13 over F4[X]/(X^2) who´s minimum homogenous weight is 37.

Homogenous weight enumerator: w(x)=1x^0+48x^37+108x^38+9x^40+48x^41+36x^42+6x^44

The gray image is a linear code over GF(4) with n=52, k=4 and d=37.
As d=37 is an upper bound for linear (52,4,4)-codes, this code is optimal over F4[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in -3.62e-008 seconds.