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

generates a code of length 71 over Z7[X]/(X^2) who´s minimum homogenous weight is 406.

Homogenous weight enumerator: w(x)=1x^0+456x^406+468x^413+294x^414+360x^420+3528x^421+390x^427+10584x^428+150x^434+168x^441+144x^448+84x^455+72x^462+42x^469+30x^476+24x^483+6x^490+6x^497

The gray image is a linear code over GF(7) with n=497, k=5 and d=406.
This code was found by Heurico 1.16 in 95.6 seconds.