The generator matrix

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

generates a code of length 88 over Z2[X]/(X^4) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+56x^82+133x^84+348x^86+1040x^88+304x^90+74x^92+44x^94+31x^96+16x^98+1x^164

The gray image is a linear code over GF(2) with n=704, k=11 and d=328.
This code was found by Heurico 1.16 in 10.5 seconds.