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

generates a code of length 82 over Z5[X]/(X^2) who´s minimum homogenous weight is 305.

Homogenous weight enumerator: w(x)=1x^0+176x^305+416x^310+100x^313+484x^315+600x^318+356x^320+2600x^323+324x^325+5100x^328+316x^330+4100x^333+240x^335+188x^340+152x^345+112x^350+124x^355+76x^360+88x^365+32x^370+28x^375+4x^380+8x^385

The gray image is a linear code over GF(5) with n=410, k=6 and d=305.
This code was found by Heurico 1.16 in 1.48 seconds.