The generator matrix

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

generates a code of length 62 over Z2[X]/(X^4) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+319x^56+1080x^57+2166x^58+2946x^59+3516x^60+4326x^61+4437x^62+4376x^63+3546x^64+2712x^65+1682x^66+820x^67+464x^68+186x^69+102x^70+44x^71+16x^72+16x^73+4x^74+6x^75+2x^76+1x^78

The gray image is a linear code over GF(2) with n=496, k=15 and d=224.
This code was found by Heurico 1.16 in 12.8 seconds.