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

generates a code of length 64 over Z7[X]/(X^2) who´s minimum homogenous weight is 343.

Homogenous weight enumerator: w(x)=1x^0+78x^343+768x^350+1446x^357+588x^362+1656x^364+3528x^369+1734x^371+16758x^376+1938x^378+39984x^383+2052x^385+39984x^390+2040x^392+1956x^399+1416x^406+966x^413+522x^420+204x^427+18x^434+12x^441

The gray image is a linear code over GF(7) with n=448, k=6 and d=343.
This code was found by Heurico 1.16 in 14.5 seconds.