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

generates a code of length 78 over Z7[X]/(X^2) who´s minimum homogenous weight is 427.

Homogenous weight enumerator: w(x)=1x^0+240x^427+1038x^434+1440x^441+1626x^448+294x^450+1842x^455+5292x^457+1860x^462+31752x^464+1830x^469+63504x^471+1584x^476+1524x^483+1452x^490+1158x^497+672x^504+342x^511+144x^518+48x^525+6x^532

The gray image is a linear code over GF(7) with n=546, k=6 and d=427.
This code was found by Heurico 1.16 in 17.8 seconds.