The generator matrix

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

generates a code of length 10 over F4[X,sigma]/(X^2) who�s minimum homogenous weight is 28.

Homogenous weight enumerator: w(x)=1x^0+120x^28+135x^32

The gray image is a linear code over GF(4) with n=40, k=4 and d=28.
As d=28 is an upper bound for linear (40,4,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0 seconds.