The generator matrix

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

generates a code of length 7 over Z7[X]/(X^2) who´s minimum homogenous weight is 28.

Homogenous weight enumerator: w(x)=1x^0+210x^28+378x^35+15414x^42+804x^49

The gray image is a linear code over GF(7) with n=49, k=5 and d=28.
This code was found by Heurico 1.16 in 0.00954 seconds.