Best Known (87, 87+60, s)-Nets in Base 2
(87, 87+60, 60)-Net over F2 — Constructive and digital
Digital (87, 147, 60)-net over F2, using
- 1 times m-reduction [i] based on digital (87, 148, 60)-net over F2, using
- trace code for nets [i] based on digital (13, 74, 30)-net over F4, using
- net from sequence [i] based on digital (13, 29)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 13 and N(F) ≥ 30, using
- F4 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 13 and N(F) ≥ 30, using
- net from sequence [i] based on digital (13, 29)-sequence over F4, using
- trace code for nets [i] based on digital (13, 74, 30)-net over F4, using
(87, 87+60, 68)-Net over F2 — Digital
Digital (87, 147, 68)-net over F2, using
(87, 87+60, 298)-Net in Base 2 — Upper bound on s
There is no (87, 147, 299)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(2147, 299, S2, 60), but
- adding a parity check bit [i] would yield OA(2148, 300, S2, 61), but
- the linear programming bound shows that M ≥ 105136 800473 490952 225520 267805 459539 798214 028504 347707 095150 350686 812959 234065 848612 773808 595863 593519 982303 777658 516010 091519 767770 474919 303087 529165 787729 563072 888056 116502 296296 071562 989888 891101 634501 491255 596641 166293 581253 564296 028466 674899 811761 448707 686400 / 225 798485 420294 252360 409611 183444 597041 002675 517141 633160 686125 901188 822156 048472 668432 443090 701115 504331 078053 724134 453816 241595 495681 917850 232579 046224 872692 104555 517294 977093 227439 954041 075655 206185 993731 221251 204789 > 2148 [i]
- adding a parity check bit [i] would yield OA(2148, 300, S2, 61), but