The generator matrix

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

generates a code of length 20 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 60.

Homogenous weight enumerator: w(x)=1x^0+240x^60+15x^64

The gray image is a linear code over GF(4) with n=80, k=4 and d=60.
As d=60 is an upper bound for linear (80,4,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 3.24e-008 seconds.