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  X  X  1  1  1  1  1  1
 0  X  0  0  X  0  X  X a^2*X a^2*X a^2*X a^2*X a^2*X a^2*X  0  0  X  X  0  0  X  X a^2*X a^2*X  0  X a*X a^2*X  0  X a*X a^2*X  0  X  X  X a*X a^2*X  X a*X  0
 0  0  X  0 a^2*X  X a^2*X  0 a^2*X a^2*X  X  X a^2*X  X  0  X a^2*X  0  0  X a^2*X  0 a^2*X  X a*X a*X a*X a*X a*X a*X a*X a*X a*X  0  0  X a^2*X  0  X a^2*X a*X
 0  0  0  X  X  X a*X a^2*X  0  X  0  X a*X a*X a*X a^2*X  0 a*X a^2*X a*X a^2*X  X a^2*X a^2*X  0 a^2*X  0 a^2*X  X  X a^2*X a*X a*X a*X  0  0 a^2*X  0 a*X a*X  0

generates a code of length 41 over F4[X]/(X^2) who´s minimum homogenous weight is 116.

Homogenous weight enumerator: w(x)=1x^0+66x^116+48x^117+75x^120+288x^121+51x^124+432x^125+27x^128+12x^132+6x^136+6x^140+6x^148+3x^152+3x^156

The gray image is a linear code over GF(4) with n=164, k=5 and d=116.
This code was found by Heurico 1.16 in 0.015 seconds.