The generator matrix

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

generates a code of length 61 over Z2[X]/(X^4) who´s minimum homogenous weight is 54.

Homogenous weight enumerator: w(x)=1x^0+98x^54+1030x^55+2093x^56+3754x^57+5444x^58+7360x^59+8620x^60+9196x^61+8670x^62+7078x^63+5072x^64+3904x^65+1829x^66+798x^67+388x^68+132x^69+30x^70+20x^71+6x^73+9x^74+2x^75+2x^76

The gray image is a linear code over GF(2) with n=488, k=16 and d=216.
This code was found by Heurico 1.16 in 33.9 seconds.