The generator matrix

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

generates a code of length 44 over F4[X]/(X^2) who´s minimum homogenous weight is 124.

Homogenous weight enumerator: w(x)=1x^0+558x^124+1089x^128+915x^132+939x^136+528x^140+48x^144+3x^148+3x^152+12x^156

The gray image is a linear code over GF(4) with n=176, k=6 and d=124.
This code was found by Heurico 1.16 in 0.0931 seconds.