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

generates a code of length 59 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 176.

Homogenous weight enumerator: w(x)=1x^0+39x^176+192x^177+12x^180+9x^188+3x^204

The gray image is a linear code over GF(4) with n=236, k=4 and d=176.
As d=176 is an upper bound for linear (236,4,4)-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.