The generator matrix

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

generates a code of length 69 over Z2[X]/(X^4) who´s minimum homogenous weight is 62.

Homogenous weight enumerator: w(x)=1x^0+188x^62+1424x^63+2308x^64+4158x^65+5453x^66+7164x^67+8095x^68+8778x^69+7647x^70+7144x^71+5196x^72+3898x^73+2136x^74+1224x^75+379x^76+184x^77+63x^78+48x^79+21x^80+20x^81+1x^82+4x^83+2x^85

The gray image is a linear code over GF(2) with n=552, k=16 and d=248.
This code was found by Heurico 1.16 in 39 seconds.