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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+38x^80+54x^81+55x^82+62x^83+196x^84+160x^85+979x^86+128x^87+188x^88+64x^89+36x^90+24x^91+20x^92+8x^93+12x^94+8x^95+5x^96+2x^97+4x^98+2x^99+1x^102+1x^162

The gray image is a linear code over GF(2) with n=688, k=11 and d=320.
This code was found by Heurico 1.16 in 0.844 seconds.