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

generates a code of length 68 over Z7[X]/(X^2) who´s minimum homogenous weight is 371.

Homogenous weight enumerator: w(x)=1x^0+456x^371+1224x^378+1680x^385+294x^390+1728x^392+5292x^397+1854x^399+31752x^404+2010x^406+63504x^411+1788x^413+1980x^420+1656x^427+1218x^434+690x^441+366x^448+102x^455+48x^462+6x^469

The gray image is a linear code over GF(7) with n=476, k=6 and d=371.
This code was found by Heurico 1.16 in 15.2 seconds.