The generator matrix

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

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+432x^48+432x^52+144x^56+15x^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 6.87e-008 seconds.