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

generates a code of length 56 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 168.

Homogenous weight enumerator: w(x)=1x^0+240x^168+12x^176+3x^192

The gray image is a linear code over GF(4) with n=224, k=4 and d=168.
As d=168 is an upper bound for linear (224,4,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.032 seconds.