The generator matrix

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

generates a code of length 88 over Z2[X]/(X^4) who´s minimum homogenous weight is 81.

Homogenous weight enumerator: w(x)=1x^0+590x^81+1912x^82+3304x^83+4454x^84+5916x^85+6145x^86+7238x^87+7478x^88+7142x^89+6485x^90+5026x^91+3671x^92+2908x^93+1592x^94+842x^95+403x^96+264x^97+81x^98+32x^99+25x^100+12x^101+9x^102+4x^103+2x^107

The gray image is a linear code over GF(2) with n=704, k=16 and d=324.
This code was found by Heurico 1.16 in 57 seconds.