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

generates a code of length 47 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 140.

Homogenous weight enumerator: w(x)=1x^0+45x^140+192x^141+15x^144+3x^188

The gray image is a linear code over GF(4) with n=188, k=4 and d=140.
As d=140 is an upper bound for linear (188,4,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.015 seconds.