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

generates a code of length 70 over Z7[X]/(X^2) who´s minimum homogenous weight is 378.

Homogenous weight enumerator: w(x)=1x^0+192x^378+696x^385+1440x^392+42x^396+1548x^399+1008x^403+1806x^406+9072x^410+1872x^413+36288x^417+1830x^420+54432x^424+1884x^427+1800x^434+1482x^441+1074x^448+816x^455+288x^462+78x^469

The gray image is a linear code over GF(7) with n=490, k=6 and d=378.
This code was found by Heurico 1.16 in 15.6 seconds.