The generator matrix

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

generates a code of length 20 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+351x^56+312x^60+357x^64+3x^72

The gray image is a linear code over GF(4) with n=80, k=5 and d=56.
This code was found by Heurico 1.16 in 0.141 seconds.