The generator matrix

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

generates a code of length 44 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 41.

Homogenous weight enumerator: w(x)=1x^0+470x^41+650x^42+874x^43+637x^44+518x^45+376x^46+262x^47+96x^48+148x^49+29x^50+24x^51+1x^52+8x^53+1x^56+1x^58

The gray image is a code over GF(2) with n=352, k=12 and d=164.
This code was found by Heurico 1.16 in 0.375 seconds.