Best Known (109, 109+57, s)-Nets in Base 2
(109, 109+57, 68)-Net over F2 — Constructive and digital
Digital (109, 166, 68)-net over F2, using
- 10 times m-reduction [i] based on digital (109, 176, 68)-net over F2, using
- trace code for nets [i] based on digital (21, 88, 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, 88, 34)-net over F4, using
(109, 109+57, 72)-Net in Base 2 — Constructive
(109, 166, 72)-net in base 2, using
- trace code for nets [i] based on (26, 83, 36)-net in base 4, using
- net from sequence [i] based on (26, 35)-sequence in base 4, using
- base expansion [i] based on digital (52, 35)-sequence over F2, using
- t-expansion [i] based on digital (51, 35)-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 3 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]
- t-expansion [i] based on digital (51, 35)-sequence over F2, using
- base expansion [i] based on digital (52, 35)-sequence over F2, using
- net from sequence [i] based on (26, 35)-sequence in base 4, using
(109, 109+57, 112)-Net over F2 — Digital
Digital (109, 166, 112)-net over F2, using
(109, 109+57, 630)-Net in Base 2 — Upper bound on s
There is no (109, 166, 631)-net in base 2, because
- 1 times m-reduction [i] would yield (109, 165, 631)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 46 852056 229336 873618 948987 469289 918214 647690 073556 > 2165 [i]