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

generates a code of length 69 over Z7[X]/(X^2) who´s minimum homogenous weight is 378.

Homogenous weight enumerator: w(x)=1x^0+642x^378+1290x^385+1608x^392+1836x^399+2058x^402+1902x^406+24696x^409+1902x^413+74088x^416+2004x^420+1866x^427+1470x^434+1020x^441+732x^448+444x^455+54x^462+24x^469+12x^476

The gray image is a linear code over GF(7) with n=483, k=6 and d=378.
This code was found by Heurico 1.16 in 34.5 seconds.