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

generates a code of length 39 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 116.

Homogenous weight enumerator: w(x)=1x^0+36x^116+192x^117+12x^120+12x^124+3x^128

The gray image is a linear code over GF(4) with n=156, k=4 and d=116.
As d=116 is an upper bound for linear (156,4,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in -1.01e-007 seconds.