The generator matrix

 1  0  1  1  1  1  1  1  1  0  1  1  1  0  1  1  1  1  1  X  1  1 a*X  1  1  1  1  1  0  1  1  1  1  1 a*X  1 a*X  1  1  1  1  1 a^2*X  1  1  1 a*X  1  1  X  1  1  X  1  1  1  1  1  1  1 a*X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  0  1  1  1  1  1  1  1
 0  1  1  a a^2  0 a^2*X+1 a^2*X+a^2  a  1  0 a^2*X+1  a  1 a^2*X+a^2  X a^2*X+a^2 X+a a^2*X+1  1 a^2*X+a^2  a  1  1 a*X+a a*X+a^2  0 X+a  1 a^2*X+a^2 a*X a^2*X+1 X+a  X  1 a^2*X+1  1 a^2*X+a a*X+1  X a*X a*X+1  1 X+1 a*X+a^2 X+1  1 a^2 X+a  1 a^2*X+a^2  1  1 a^2*X  X  a X+a^2 a^2*X a^2*X a^2*X+1  1 X+a^2  1  a  X a*X+1  a a*X+1 X+1  X X+1  0  0 a^2*X+1 a^2*X+1  0 X+a a^2  1  1  1 a^2*X+a^2 X+a^2 a^2*X+a a^2*X a*X  a  X
 0  0 a^2*X  0  0  0  X  X  X  X  X  X a^2*X a^2*X a*X  X a*X a^2*X a^2*X  0 a^2*X a^2*X a*X  X  0 a*X  0 a*X  X  0 a^2*X a*X a^2*X a^2*X  X a*X a^2*X a*X a*X  0 a^2*X  0 a^2*X a^2*X a^2*X  0 a^2*X a*X  0 a*X  X a*X  0 a^2*X a^2*X a*X a^2*X  0 a*X  X  0 a*X a*X a^2*X a*X  0  0 a^2*X  0  0  X  0 a*X  X a*X  X  0 a^2*X a^2*X  X  0  X a*X a^2*X  X a*X  X a*X
 0  0  0  X  0  X a^2*X  0  X a^2*X  X  0 a*X a^2*X  0 a^2*X  0  0 a*X a*X  X  X a^2*X  0  X  0 a*X a^2*X a^2*X  0  X a*X  0  X  0 a^2*X a*X a^2*X  0  0 a^2*X  X  X  0 a^2*X a^2*X a^2*X  X a^2*X a*X  X  X  X  0 a^2*X  0 a*X a*X  X a^2*X  0  X  0 a^2*X a*X a^2*X  0 a*X a*X a^2*X a*X a*X  X  0 a*X  X a^2*X a^2*X  0  X a*X a^2*X a*X  0 a^2*X a^2*X a*X a^2*X
 0  0  0  0 a^2*X a^2*X  X a^2*X a*X  0 a^2*X  X  X a*X  X a*X a*X  X a^2*X a^2*X  0 a^2*X a^2*X a*X a*X a^2*X  0 a^2*X a^2*X a*X a*X  0 a^2*X  X a^2*X  X  X a*X  0 a^2*X  X a*X a*X a*X a*X a*X a^2*X a*X a^2*X a^2*X  0  0  0 a^2*X  0  0  0  X a^2*X a^2*X a*X  0 a*X a^2*X a*X  0 a*X a*X  0  X a^2*X a^2*X a*X a^2*X a^2*X a*X  0 a^2*X a^2*X  0  X  0  0  0  0  X a^2*X a^2*X

generates a code of length 88 over F4[X]/(X^2) who´s minimum homogenous weight is 248.

Homogenous weight enumerator: w(x)=1x^0+108x^248+72x^249+228x^250+204x^251+357x^252+420x^253+780x^254+372x^255+651x^256+516x^257+1020x^258+396x^259+657x^260+324x^261+1032x^262+576x^263+705x^264+600x^265+1056x^266+804x^267+789x^268+492x^269+1116x^270+468x^271+393x^272+372x^273+684x^274+228x^275+240x^276+204x^277+144x^278+24x^279+72x^280+72x^281+84x^282+27x^284+15x^288+18x^292+30x^296+12x^300+9x^308+9x^312+3x^316

The gray image is a linear code over GF(4) with n=352, k=7 and d=248.
This code was found by Heurico 1.16 in 2.08 seconds.