The generator matrix

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

generates a code of length 16 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+1440x^44+1212x^48+1440x^52+3x^64

The gray image is a linear code over GF(4) with n=64, k=6 and d=44.
As d=44 is an upper bound for linear (64,6,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 6.
This code was found by Heurico 1.16 in 2.92 seconds.