The generator matrix

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

generates a code of length 16 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 48.

Homogenous weight enumerator: w(x)=1x^0+252x^48+3x^64

The gray image is a linear code over GF(4) with n=64, k=4 and d=48.
As d=48 is an upper bound for linear (64,4,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0 seconds.