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  2  X  X  0  X  X  0  1  0  0  1  1  1  1  1  2  1  2  0  1  2  1  2  1  1  0  0  1  X  1  2  2  1  1  X  2  1  0  X  0
 0  X  0  0  0  0  0  0  2  2  X X+2  X  0  0  2 X+2 X+2  X  X  X  X  0  X X+2  X  0  X X+2  2 X+2  2  2 X+2  X  0  0  X  2  X  2  2 X+2  2  0  X  2  X  0  X X+2  0  2  X  X  0  2  2  0  2  2 X+2  2  0  2  X  0  X  2  X X+2  X  X  X  2  X X+2  X
 0  0  X  0  0  0  0  0  0  0  0  0  2 X+2 X+2 X+2  X X+2 X+2  X  2  2 X+2 X+2  0  0  X  X  X X+2 X+2 X+2  2  2 X+2  X  X  2  X X+2  X  0  0  X  2  0  2  X  0  2 X+2  0  2 X+2  0  2  X X+2  X X+2  X  0  2  2  2  X  X  X  2  2  X  0  0  2  X  0  2  0
 0  0  0  X  0  0  2 X+2  X  X  X  X  2 X+2  X  2  2  0  2  2  2  2  2  X X+2  X  2  X X+2 X+2  X X+2  0  2  0  0  0 X+2  0  0 X+2  X  X  0  0  X  2  0  0  0 X+2  X X+2  X  0  X  0 X+2  2  X  2 X+2  0  0  X  0  2  2  0  2  2  0 X+2 X+2  X  0  2  X
 0  0  0  0  X  0 X+2 X+2  X  2 X+2 X+2  0  X  X  0  2  X  0 X+2 X+2  X X+2  X  2  2  X  2  2  0 X+2  2  0 X+2  X X+2  2  0 X+2  0 X+2  0  0  0  X  X  X X+2  0  0  0  0  0 X+2  2 X+2  2 X+2  0  2  2  2 X+2  0  X X+2 X+2  X  X  2 X+2 X+2  0  X  X  0  2  2
 0  0  0  0  0  X  X  2 X+2  X X+2  2  X  X  0  X  0 X+2 X+2  0  X  2  2 X+2  2  X X+2 X+2  2  X  2  2 X+2  0  X X+2  0  0  2  2  0  2 X+2  X  2  0  X  0  2  0  2  X  0  0 X+2  X  0  X  0  2 X+2  0 X+2  X  0 X+2 X+2  2  0  0  X  X  0  2 X+2  2  X  X

generates a code of length 78 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+186x^68+12x^69+394x^70+68x^71+607x^72+188x^73+720x^74+304x^75+765x^76+472x^77+976x^78+444x^79+768x^80+304x^81+662x^82+172x^83+419x^84+44x^85+296x^86+32x^87+178x^88+4x^89+100x^90+4x^91+45x^92+18x^94+6x^96+2x^98+1x^108

The gray image is a code over GF(2) with n=312, k=13 and d=136.
This code was found by Heurico 1.16 in 7.71 seconds.