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

generates a code of length 74 over Z7[X]/(X^2) who´s minimum homogenous weight is 420.

Homogenous weight enumerator: w(x)=1x^0+180x^420+438x^427+504x^434+372x^441+14406x^444+222x^448+204x^455+132x^462+84x^469+102x^476+60x^483+60x^490+30x^497+6x^504+6x^518

The gray image is a linear code over GF(7) with n=518, k=5 and d=420.
This code was found by Heurico 1.16 in 0.406 seconds.