The generator matrix

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

generates a code of length 44 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 39.

Homogenous weight enumerator: w(x)=1x^0+72x^39+231x^40+274x^41+533x^42+632x^43+847x^44+504x^45+478x^46+174x^47+138x^48+86x^49+53x^50+48x^51+20x^52+2x^55+2x^56+1x^68

The gray image is a code over GF(2) with n=352, k=12 and d=156.
This code was found by Heurico 1.16 in 0.25 seconds.