The generator matrix

 1  0  1  1  1 X+2  1  1  0 X+2  1  1  1  1  2  1  X  1  1  1  2  X  1  1  1 X+2  1  0  1  1  0  1  0  1  1  1  0  1  1  1  X  1  2  2  1  1  2  1  1 X+2 X+2  1  1  1  1  1  1  1  1  1  1  0  1  1 X+2  1  0  0  1  1  1  0  X  1  X  X  2  0
 0  1  1  0  1  1  X X+3  1  1 X+3 X+2  1  2  1  0  1  3  1  X  1  1  X X+1 X+3  1  1  1  0  X  1  0  1  X  1 X+3  1 X+2 X+1  0  1  0  1  1 X+2  2  1  3  1  1  1  X  1 X+3  3 X+2  3  0 X+1  0 X+1  1 X+2 X+3  1  3  1  1  1  2  X  X  1  3 X+2  X  X  1
 0  0  X  0  0  0  0  0  0  2  0  0  2  0  2  0  2  0  2  2  0  2  2  X  X X+2 X+2  X X+2 X+2  X X+2 X+2 X+2  X  X  X X+2  X  X  2  X  X X+2  2  2  X X+2  X  X  0 X+2  0  0  0 X+2  X X+2  2  2  0  X  0  X  2 X+2  0  0 X+2  0 X+2 X+2  X  2  2  0 X+2  X
 0  0  0  X  0  0  X  2  X X+2  X  2 X+2  0  X X+2  2  X  2  2  2 X+2 X+2  0  0  2  0  0  0 X+2 X+2 X+2  X X+2 X+2 X+2  0  X  X  2  X  0  2  0  2 X+2  X  X  0  X  2  0  X  0  2  X X+2  2  X  2  0  X  X  2  0  X  2 X+2 X+2  0  2  2  X X+2  X  X X+2 X+2
 0  0  0  0  X  0  0  X  X X+2  2  2  2  X  2  X X+2 X+2  2 X+2 X+2  0  X X+2  2  X X+2  0 X+2 X+2  2  2  X X+2 X+2  X X+2  2  2  0 X+2 X+2  2  X  0 X+2  X  0  2  0  0  0 X+2  2  X  X  0 X+2  0  0  0 X+2  0 X+2 X+2 X+2 X+2  2  X X+2  X  0  2  0 X+2  X  2  0
 0  0  0  0  0  2  2  2  2  0  2  2  0  2  2  2  2  2  2  0  0  0  0  0  2  0  2  2  0  2  2  0  2  0  2  0  2  2  0  2  2  2  0  2  0  2  0  2  2  0  0  0  0  0  0  2  0  2  2  2  2  2  0  2  2  2  2  2  0  2  2  2  2  0  2  0  2  2

generates a code of length 78 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+86x^69+158x^70+282x^71+326x^72+440x^73+639x^74+630x^75+625x^76+654x^77+729x^78+656x^79+671x^80+564x^81+441x^82+456x^83+249x^84+190x^85+139x^86+58x^87+62x^88+40x^89+30x^90+26x^91+15x^92+6x^93+6x^94+4x^95+4x^97+2x^98+3x^100

The gray image is a code over GF(2) with n=312, k=13 and d=138.
This code was found by Heurico 1.16 in 5.68 seconds.