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

generates a code of length 53 over Z7[X]/(X^2) who´s minimum homogenous weight is 294.

Homogenous weight enumerator: w(x)=1x^0+162x^294+402x^301+480x^308+390x^315+14406x^318+228x^322+246x^329+144x^336+150x^343+120x^350+60x^357+12x^364+6x^371

The gray image is a linear code over GF(7) with n=371, k=5 and d=294.
This code was found by Heurico 1.16 in 0.254 seconds.