The generator matrix

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

generates a code of length 97 over Z2[X]/(X^4) who´s minimum homogenous weight is 89.

Homogenous weight enumerator: w(x)=1x^0+94x^89+1026x^90+2166x^91+3300x^92+4512x^93+5446x^94+6294x^95+6917x^96+7054x^97+7157x^98+6026x^99+5080x^100+3848x^101+2576x^102+1804x^103+1111x^104+556x^105+301x^106+124x^107+82x^108+28x^109+20x^110+2x^111+5x^112+4x^113+2x^114

The gray image is a linear code over GF(2) with n=776, k=16 and d=356.
This code was found by Heurico 1.16 in 54.7 seconds.