The generator matrix

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

generates a code of length 43 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 39.

Homogenous weight enumerator: w(x)=1x^0+288x^39+797x^40+1414x^41+1040x^42+1446x^43+1106x^44+1020x^45+449x^46+356x^47+160x^48+78x^49+30x^50+6x^51+1x^54

The gray image is a code over GF(2) with n=344, k=13 and d=156.
This code was found by Heurico 1.16 in 0.406 seconds.