The generator matrix

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

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+242x^39+781x^40+1530x^41+1784x^42+2882x^43+2524x^44+2468x^45+1704x^46+1272x^47+649x^48+354x^49+86x^50+66x^51+12x^52+16x^53+8x^54+2x^55+1x^56+2x^58

The gray image is a code over GF(2) with n=352, k=14 and d=156.
This code was found by Heurico 1.16 in 1.66 seconds.