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

generates a code of length 55 over Z7[X]/(X^2) who´s minimum homogenous weight is 308.

Homogenous weight enumerator: w(x)=1x^0+192x^308+558x^315+450x^322+252x^329+14406x^330+330x^336+198x^343+144x^350+126x^357+72x^364+42x^371+30x^378+6x^385

The gray image is a linear code over GF(7) with n=385, k=5 and d=308.
This code was found by Heurico 1.16 in 0.267 seconds.