The generator matrix

 1  0  0  1  1  1 3X  1  1  1  1  2  0 2X  0  X  1  1  X  1  1  1 2X+2 2X+2  1  X  1  X  0 3X+2  X  1  1  1  1 X+2  1  1  1  1  0  1 2X+2  1
 0  1  0  0  1 X+1  1 2X X+1 3X+1 3X+2  1  1 2X+2 X+2  1 X+3 3X+1  1 2X+2 X+1  X 3X+2  1  2  0 3X+2  0  1  1  1 2X  X  1 X+2  1 2X+1 X+2  0 X+3  1 2X+1  1  0
 0  0  1  1  1  X X+1  X 3X+2 2X+1 X+1 X+2  1  1  1 3X+2 2X+1  2 2X+1 3X+1 X+1 X+3  1  1 2X  1 3X  1 2X+2 2X+1 2X+3 X+2  0 2X  3 3X+3 X+3 3X+2 X+1 X+1 X+2  1 2X  0
 0  0  0  X 2X 3X X+2 3X+2  0 3X+2  0 3X 2X+2 X+2 3X 2X+2 2X 2X+2  2 2X X+2  X 2X+2 X+2  X 3X+2 2X 2X+2 3X+2  0 3X  2  0 2X+2 3X  0  2 3X+2 2X+2 3X  2 3X+2  X 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+78x^38+570x^39+1578x^40+2580x^41+3885x^42+4898x^43+5729x^44+4830x^45+4262x^46+2300x^47+1105x^48+586x^49+219x^50+82x^51+33x^52+18x^53+2x^54+6x^55+2x^56+2x^57+2x^58

The gray image is a code over GF(2) with n=352, k=15 and d=152.
This code was found by Heurico 1.16 in 6.59 seconds.