The generator matrix

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

generates a code of length 44 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 37.

Homogenous weight enumerator: w(x)=1x^0+136x^37+923x^38+2610x^39+5229x^40+9926x^41+14906x^42+20342x^43+22081x^44+21354x^45+15698x^46+9464x^47+4654x^48+2426x^49+946x^50+248x^51+63x^52+46x^53+7x^54+6x^55+4x^56+2x^59

The gray image is a code over GF(2) with n=352, k=17 and d=148.
This code was found by Heurico 1.16 in 86.9 seconds.