The generator matrix

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

generates a code of length 44 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+330x^40+756x^41+1341x^42+1206x^43+1397x^44+1028x^45+1027x^46+454x^47+367x^48+178x^49+63x^50+20x^51+17x^52+6x^53+1x^54

The gray image is a code over GF(2) with n=352, k=13 and d=160.
This code was found by Heurico 1.16 in 0.438 seconds.