The generator matrix

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

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.