The generator matrix

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

generates a code of length 14 over F4[X]/(X^2) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+33x^40+192x^42+24x^44+3x^48+3x^56

The gray image is a linear code over GF(4) with n=56, k=4 and d=40.
As d=40 is an upper bound for linear (56,4,4)-codes, this code is optimal over F4[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 3.24e-008 seconds.