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

generates a code of length 81 over Z7[X]/(X^2) who´s minimum homogenous weight is 448.

Homogenous weight enumerator: w(x)=1x^0+624x^448+1260x^455+1476x^462+294x^468+1872x^469+5292x^475+1764x^476+31752x^482+1782x^483+63504x^489+1770x^490+1572x^497+1692x^504+1188x^511+840x^518+534x^525+270x^532+126x^539+36x^546

The gray image is a linear code over GF(7) with n=567, k=6 and d=448.
This code was found by Heurico 1.16 in 20.6 seconds.