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  1  1  1  1  1  1  1  1  1  1  1  1  1
 0  X  0  0  X  X a*X a^3*X  0 2*X a^2*X  X a^2*X a*X a^2*X a^5*X a*X  X a*X 2*X a^6*X a^2*X  0 a*X  0  X 2*X a^6*X a^2*X 2*X a^2*X a^3*X a^5*X a^6*X a^2*X 2*X a^3*X  X  0 a^6*X 2*X a^7*X  X 2*X  0  X a^6*X a^6*X a^7*X a^5*X a^5*X  0 a^5*X a^2*X a^7*X a^2*X a*X a^6*X 2*X a^7*X a^6*X a^3*X a^6*X a^2*X  X a^3*X 2*X a*X a*X a^7*X a^2*X 2*X a^2*X  0  X a^2*X 2*X  X a*X  0  0
 0  0  X  0 a^7*X a^6*X a*X 2*X  X a^6*X a^6*X  X a^3*X  X a^5*X  X a^2*X a^6*X a^5*X a^6*X a^5*X a^7*X a^5*X 2*X a^2*X  0  0  0 2*X 2*X a^2*X a^2*X a*X a^2*X a^5*X a^2*X a^6*X a*X a^3*X a^7*X a^7*X  0  X a*X a*X a^2*X 2*X 2*X a^2*X a^3*X a^6*X a*X  X a^7*X a^7*X  0 a*X a^7*X a^5*X a^5*X  0 a^7*X a^3*X a*X  X a^6*X a*X  0 a*X  0  X a^5*X 2*X a^2*X a*X a*X a^2*X a^3*X a^6*X a*X  X
 0  0  0  X a^7*X  X a^5*X a*X a*X a^7*X 2*X a^7*X a^3*X a^2*X 2*X a^3*X a^6*X 2*X  X a^6*X a^6*X a*X 2*X  0  X a^6*X a^5*X a^6*X  0 2*X a^3*X a*X a^3*X a^3*X a*X  0  X a^5*X a^6*X a*X a^5*X 2*X  0 2*X a*X a^5*X  0 2*X a*X a^7*X  0  0 a*X a^5*X a^3*X a^2*X 2*X  X a^7*X a^7*X a*X a^7*X a^3*X 2*X a^3*X  0 a^2*X a^3*X a^6*X a^3*X a*X  0 a^2*X a^7*X 2*X a^3*X a^5*X a^3*X  0 a^7*X a^7*X

generates a code of length 81 over F9[X]/(X^2) who´s minimum homogenous weight is 612.

Homogenous weight enumerator: w(x)=1x^0+264x^612+952x^621+1128x^630+992x^639+53304x^648+576x^657+544x^666+424x^675+384x^684+240x^693+160x^702+48x^711+24x^720+8x^729

The gray image is a linear code over GF(9) with n=729, k=5 and d=612.
This code was found by Heurico 1.16 in 1.77 seconds.