The generator matrix

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

generates a code of length 38 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 36.

Homogenous weight enumerator: w(x)=1x^0+22x^36+212x^38+15x^40+4x^42+2x^56

The gray image is a code over GF(2) with n=304, k=8 and d=144.
This code was found by Heurico 1.16 in 0.032 seconds.