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

generates a code of length 79 over Z7[X]/(X^2) who´s minimum homogenous weight is 434.

Homogenous weight enumerator: w(x)=1x^0+288x^434+1086x^441+1524x^448+1704x^455+294x^456+1914x^462+5292x^463+1770x^469+31752x^470+1764x^476+63504x^477+1668x^483+1656x^490+1320x^497+1008x^504+618x^511+288x^518+156x^525+36x^532+6x^546

The gray image is a linear code over GF(7) with n=553, k=6 and d=434.
This code was found by Heurico 1.16 in 18.6 seconds.