The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  0  X  2  X 2X  X 2X+2  1  1  1  1  X  X  X  X  1  1  1  1 2X+2 2X+2  0 2X  1  1  1  1
 0  X  2 3X+2 2X X+2 2X+2 3X  0 3X+2  2 3X 2X X+2 2X+2  X 3X+2  X 3X  X X+2  X  X  X  0  2 2X 2X+2  0  2 2X 2X+2 3X+2 3X X+2  X  2 2X+2 2X+2 2X+2  0 2X 3X+2 X+2

generates a code of length 44 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+118x^44+4x^46+4x^50+1x^56

The gray image is a code over GF(2) with n=352, k=7 and d=176.
This code was found by Heurico 1.16 in 0.032 seconds.