The generator matrix

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

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+204x^176+24x^180+12x^184+15x^192

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 0.109 seconds.