The generator matrix

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

generates a code of length 44 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 131.

Homogenous weight enumerator: w(x)=1x^0+144x^131+54x^132+48x^135+3x^144+6x^148

The gray image is a linear code over GF(4) with n=176, k=4 and d=131.
As d=131 is an upper bound for linear (176,4,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 4.02 seconds.