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

generates a code of length 68 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 58.

Homogenous weight enumerator: w(x)=1x^0+108x^58+391x^60+24x^61+532x^62+140x^63+748x^64+376x^65+906x^66+484x^67+953x^68+520x^69+838x^70+308x^71+704x^72+168x^73+436x^74+28x^75+237x^76+164x^78+91x^80+22x^82+10x^84+2x^86+1x^100

The gray image is a code over GF(2) with n=272, k=13 and d=116.
This code was found by Heurico 1.16 in 6.04 seconds.