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

generates a code of length 67 over Z7[X]/(X^2) who´s minimum homogenous weight is 378.

Homogenous weight enumerator: w(x)=1x^0+144x^378+480x^385+468x^392+2058x^396+378x^399+12348x^403+246x^406+180x^413+162x^420+126x^427+78x^434+60x^441+36x^448+30x^455+12x^462

The gray image is a linear code over GF(7) with n=469, k=5 and d=378.
This code was found by Heurico 1.16 in 0.347 seconds.