The generator matrix

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

generates a code of length 44 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+382x^40+96x^41+780x^42+416x^43+943x^44+416x^45+588x^46+96x^47+234x^48+92x^50+36x^52+12x^54+3x^56+1x^68

The gray image is a code over GF(2) with n=352, k=12 and d=160.
This code was found by Heurico 1.16 in 53.4 seconds.