The generator matrix

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

generates a code of length 43 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 38.

Homogenous weight enumerator: w(x)=1x^0+142x^38+718x^39+1739x^40+1730x^41+2618x^42+2634x^43+2824x^44+1660x^45+1290x^46+588x^47+267x^48+58x^49+78x^50+26x^51+8x^53+2x^55+1x^56

The gray image is a code over GF(2) with n=344, k=14 and d=152.
This code was found by Heurico 1.16 in 1.66 seconds.