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

generates a code of length 62 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 184.

Homogenous weight enumerator: w(x)=1x^0+141x^184+768x^186+96x^188+15x^192+3x^248

The gray image is a linear code over GF(4) with n=248, k=5 and d=184.
As d=184 is an upper bound for linear (248,5,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.047 seconds.