The generator matrix

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

generates a code of length 17 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 48.

Homogenous weight enumerator: w(x)=1x^0+396x^48+576x^52+51x^64

The gray image is a linear code over GF(4) with n=68, k=5 and d=48.
As d=48 is an upper bound for linear (68,5,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.062 seconds.