The generator matrix

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

generates a code of length 16 over Z7[X]/(X^2) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+120x^77+450x^84+444x^91+14406x^96+528x^98+684x^105+174x^112

The gray image is a linear code over GF(7) with n=112, k=5 and d=77.
This code was found by Heurico 1.16 in 0.0525 seconds.