The generator matrix

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

generates a code of length 40 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+140x^34+1108x^35+2726x^36+4830x^37+7845x^38+10056x^39+11805x^40+10502x^41+8204x^42+4766x^43+2204x^44+858x^45+301x^46+116x^47+48x^48+18x^49+6x^50+2x^51

The gray image is a code over GF(2) with n=320, k=16 and d=136.
This code was found by Heurico 1.16 in 20.7 seconds.