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

generates a code of length 38 over Z2[X]/(X^4) who´s minimum homogenous weight is 35.

Homogenous weight enumerator: w(x)=1x^0+64x^35+142x^36+128x^37+480x^38+142x^40+64x^43+2x^44+1x^64

The gray image is a linear code over GF(2) with n=304, k=10 and d=140.
This code was found by Heurico 1.16 in 32.8 seconds.