The generator matrix

 1  0  1  1  1  1  1  1  1  X  1  1  1  1
 0  1 (a+1)X+1  a (a+1)X+a+1  X aX+1 X+a aX+a+1  1 aX X+1 aX+a (a+1)X

generates a code of length 14 over F4[X,sigma]/(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,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 3.24e-008 seconds.