The generator matrix

 1  0  1  1  1  1  1  1  1  X  1  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 a^2*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+51x^40+72x^41+48x^42+54x^44+24x^45+6x^48

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 -3.62e-008 seconds.