The generator matrix

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

generates a code of length 48 over Z2[X]/(X^4) who´s minimum homogenous weight is 42.

Homogenous weight enumerator: w(x)=1x^0+310x^42+1774x^43+2991x^44+5512x^45+7049x^46+10424x^47+9518x^48+10478x^49+7273x^50+5486x^51+2531x^52+1388x^53+499x^54+228x^55+29x^56+30x^57+5x^58+8x^59+2x^60

The gray image is a linear code over GF(2) with n=384, k=16 and d=168.
This code was found by Heurico 1.16 in 32.7 seconds.