The generator matrix

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

generates a code of length 44 over F4[X]/(X^2) who´s minimum homogenous weight is 131.

Homogenous weight enumerator: w(x)=1x^0+144x^131+54x^132+48x^135+3x^144+6x^148

The gray image is a linear code over GF(4) with n=176, k=4 and d=131.
As d=131 is an upper bound for linear (176,4,4)-codes, this code is optimal over F4[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 4.17 seconds.