The generator matrix

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

generates a code of length 37 over Z2[X]/(X^4) who´s minimum homogenous weight is 31.

Homogenous weight enumerator: w(x)=1x^0+222x^31+995x^32+2650x^33+4233x^34+8336x^35+10087x^36+12764x^37+9874x^38+8276x^39+4310x^40+2484x^41+893x^42+312x^43+61x^44+20x^45+8x^46+6x^47+2x^49+2x^52

The gray image is a linear code over GF(2) with n=296, k=16 and d=124.
This code was found by Heurico 1.16 in 21.8 seconds.