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  X  1  1  1  1  1  1  1  1  X  1  1  1  1  1  1  X  1  1  1  1  1  1
 0  X  0  0  0  0  0  0  X 3X  X 2X 5X 3X 4X 6X 4X 5X 6X 2X 6X 2X  X  X 4X 4X 4X 6X 2X  X 4X 4X 4X 2X 2X 4X  X 6X 3X 3X 2X 5X 3X 4X 5X  X  0 4X 2X 3X  0 6X  0 3X 2X 4X  0 5X 2X 5X 3X  X  X  0  X  0 6X  0  X 3X  X  X 2X 5X  0 3X 6X  X 6X 2X 5X 3X  0  0
 0  0  X  0  0  X  X 4X 5X 4X 4X 6X  0 3X 4X  X  0 4X  X  0 2X 4X 5X 2X 3X 6X  X 6X 5X 4X  0 2X  X  X 6X 4X  X 3X  0 2X 5X 2X 6X  X  X  X  0 4X 4X 5X 6X 3X 4X 5X 4X 5X 6X  0  0 5X 5X  X  0 2X 2X 5X 6X 2X 6X  0 2X 5X 5X  X 6X 3X 2X 2X  X 5X 5X 4X 2X  0
 0  0  0  X  0 5X 4X 3X 5X 3X 3X 3X 2X 2X  X 3X 4X 2X  0  X  0 6X 4X 6X 5X 6X 6X 6X 3X  0 5X 2X  X 3X 3X 4X 6X  X  0 3X 2X 6X  X 3X 5X  0 2X  X 2X 3X 6X 4X 6X 4X 6X 3X  X  0  X  0  X  X 3X  0 3X 4X  0 4X 6X 4X 6X  X 3X  X 3X  X  X 4X 2X 5X 2X  0 6X 3X
 0  0  0  0  X 5X  X 2X 5X  0 5X 5X 6X 5X 3X 3X  X  0  X 6X 4X 4X  X 6X  X 5X 2X 5X  0 4X 4X  X 2X 4X 3X 5X 6X  0  X 4X 6X 4X 6X 5X  X 2X 5X 2X 3X  X  X 3X  0  X 3X 4X 5X 5X 4X  X 5X 4X 3X  X  0 4X  0 4X 3X 6X  0  0 4X 4X 4X 5X 5X 6X 2X 5X 4X 5X 3X 3X

generates a code of length 84 over Z7[X]/(X^2) who´s minimum homogenous weight is 462.

Homogenous weight enumerator: w(x)=1x^0+210x^462+960x^469+1350x^476+1698x^483+294x^486+1800x^490+5292x^493+1698x^497+31752x^500+1938x^504+63504x^507+1716x^511+1500x^518+1356x^525+1020x^532+768x^539+456x^546+228x^553+84x^560+24x^567

The gray image is a linear code over GF(7) with n=588, k=6 and d=462.
This code was found by Heurico 1.16 in 19.9 seconds.