The generator matrix

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

generates a code of length 43 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 37.

Homogenous weight enumerator: w(x)=1x^0+174x^37+1304x^38+2852x^39+5420x^40+7292x^41+10371x^42+10432x^43+10920x^44+7612x^45+5076x^46+2358x^47+1187x^48+356x^49+135x^50+20x^51+16x^52+6x^53+2x^54+2x^55

The gray image is a code over GF(2) with n=344, k=16 and d=148.
This code was found by Heurico 1.16 in 20.5 seconds.