The generator matrix

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

generates a code of length 37 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+89x^34+450x^36+410x^38+28x^40+45x^42+1x^72

The gray image is a code over GF(2) with n=296, k=10 and d=136.
This code was found by Heurico 1.16 in 0.094 seconds.