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

generates a code of length 77 over Z7[X]/(X^2) who´s minimum homogenous weight is 420.

Homogenous weight enumerator: w(x)=1x^0+198x^420+864x^427+1356x^434+1662x^441+1788x^448+1854x^455+14406x^456+1836x^462+86436x^463+1728x^469+1698x^476+1302x^483+1122x^490+762x^497+426x^504+174x^511+24x^518+12x^532

The gray image is a linear code over GF(7) with n=539, k=6 and d=420.
This code was found by Heurico 1.16 in 18 seconds.