Best Known (172−61, 172, s)-Nets in Base 2
(172−61, 172, 68)-Net over F2 — Constructive and digital
Digital (111, 172, 68)-net over F2, using
- 8 times m-reduction [i] based on digital (111, 180, 68)-net over F2, using
- trace code for nets [i] based on digital (21, 90, 34)-net over F4, using
- net from sequence [i] based on digital (21, 33)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 21 and N(F) ≥ 34, using
- T5 from the second 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) = 21 and N(F) ≥ 34, using
- net from sequence [i] based on digital (21, 33)-sequence over F4, using
- trace code for nets [i] based on digital (21, 90, 34)-net over F4, using
(172−61, 172, 70)-Net in Base 2 — Constructive
(111, 172, 70)-net in base 2, using
- 2 times m-reduction [i] based on (111, 174, 70)-net in base 2, using
- trace code for nets [i] based on (24, 87, 35)-net in base 4, using
- net from sequence [i] based on (24, 34)-sequence in base 4, using
- base expansion [i] based on digital (48, 34)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
- base expansion [i] based on digital (48, 34)-sequence over F2, using
- net from sequence [i] based on (24, 34)-sequence in base 4, using
- trace code for nets [i] based on (24, 87, 35)-net in base 4, using
(172−61, 172, 107)-Net over F2 — Digital
Digital (111, 172, 107)-net over F2, using
(172−61, 172, 583)-Net in Base 2 — Upper bound on s
There is no (111, 172, 584)-net in base 2, because
- 1 times m-reduction [i] would yield (111, 171, 584)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3138 113090 779557 710209 005544 272641 307623 305941 126920 > 2171 [i]