Best Known (172−72, 172, s)-Nets in Base 2
(172−72, 172, 60)-Net over F2 — Constructive and digital
Digital (100, 172, 60)-net over F2, using
- 2 times m-reduction [i] based on digital (100, 174, 60)-net over F2, using
- trace code for nets [i] based on digital (13, 87, 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, 87, 30)-net over F4, using
(172−72, 172, 73)-Net over F2 — Digital
Digital (100, 172, 73)-net over F2, using
(172−72, 172, 298)-Net in Base 2 — Upper bound on s
There is no (100, 172, 299)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(2172, 299, S2, 72), but
- adding a parity check bit [i] would yield OA(2173, 300, S2, 73), but
- the linear programming bound shows that M ≥ 584544 688823 027303 992019 163598 045555 622337 400280 994951 338862 689959 811855 040528 082821 117754 420356 767072 183448 423051 141055 104052 613046 921671 348534 995558 126247 662793 381156 267711 356435 604715 864064 / 41 038286 307966 395039 063876 290815 988845 786829 602951 956598 132356 196127 847068 738398 927371 780621 753648 857584 870215 906243 626062 026359 862634 140625 > 2173 [i]
- adding a parity check bit [i] would yield OA(2173, 300, S2, 73), but