The generator matrix

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

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+133x^40+88x^41+232x^42+378x^43+451x^44+352x^45+199x^46+52x^47+110x^48+24x^49+16x^50+2x^51+9x^52+1x^78

The gray image is a code over GF(2) with n=352, k=11 and d=160.
This code was found by Heurico 1.16 in 0.094 seconds.