The generator matrix

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

generates a code of length 60 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 180.

Homogenous weight enumerator: w(x)=1x^0+240x^180+15x^192

The gray image is a linear code over GF(4) with n=240, k=4 and d=180.
As d=180 is an upper bound for linear (240,4,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.031 seconds.