The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 38.

Homogenous weight enumerator: w(x)=1x^0+208x^38+1250x^39+3110x^40+5312x^41+7552x^42+10030x^43+10374x^44+10512x^45+7829x^46+5106x^47+2629x^48+1046x^49+378x^50+118x^51+38x^52+24x^53+9x^54+8x^55+2x^57

The gray image is a linear code over GF(2) with n=352, k=16 and d=152.
This code was found by Heurico 1.16 in 29.7 seconds.