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

generates a code of length 82 over Z7[X]/(X^2) who´s minimum homogenous weight is 448.

Homogenous weight enumerator: w(x)=1x^0+108x^448+696x^455+1308x^462+1632x^469+294x^474+1806x^476+5292x^481+1746x^483+31752x^488+1824x^490+63504x^495+1656x^497+1518x^504+1602x^511+1104x^518+840x^525+558x^532+270x^539+108x^546+30x^553

The gray image is a linear code over GF(7) with n=574, k=6 and d=448.
This code was found by Heurico 1.16 in 19.3 seconds.