The generator matrix

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

generates a code of length 44 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 38.

Homogenous weight enumerator: w(x)=1x^0+211x^38+1474x^39+2869x^40+5198x^41+7258x^42+10358x^43+10660x^44+10700x^45+7353x^46+5324x^47+2416x^48+1078x^49+410x^50+150x^51+38x^52+16x^53+14x^54+6x^55+2x^58

The gray image is a code over GF(2) with n=352, k=16 and d=152.
This code was found by Heurico 1.16 in 22 seconds.