The generator matrix

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

generates a code of length 28 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 83.

Homogenous weight enumerator: w(x)=1x^0+144x^83+57x^84+48x^87+3x^96+3x^100

The gray image is a linear code over GF(4) with n=112, k=4 and d=83.
As d=83 is an upper bound for linear (112,4,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.094 seconds.