Best Known (106−59, 106, s)-Nets in Base 2
(106−59, 106, 34)-Net over F2 — Constructive and digital
Digital (47, 106, 34)-net over F2, using
- t-expansion [i] based on digital (45, 106, 34)-net over F2, using
- net from sequence [i] based on digital (45, 33)-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 1 place 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]
- net from sequence [i] based on digital (45, 33)-sequence over F2, using
(106−59, 106, 36)-Net over F2 — Digital
Digital (47, 106, 36)-net over F2, using
- net from sequence [i] based on digital (47, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 47 and N(F) ≥ 36, using
(106−59, 106, 102)-Net over F2 — Upper bound on s (digital)
There is no digital (47, 106, 103)-net over F2, because
- 11 times m-reduction [i] would yield digital (47, 95, 103)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(295, 103, F2, 48) (dual of [103, 8, 49]-code), but
- residual code [i] would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
- adding a parity check bit [i] would yield linear OA(248, 55, F2, 25) (dual of [55, 7, 26]-code), but
- “vT4†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(248, 55, F2, 25) (dual of [55, 7, 26]-code), but
- residual code [i] would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(295, 103, F2, 48) (dual of [103, 8, 49]-code), but
(106−59, 106, 104)-Net in Base 2 — Upper bound on s
There is no (47, 106, 105)-net in base 2, because
- 7 times m-reduction [i] would yield (47, 99, 105)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(299, 105, S2, 52), but
- adding a parity check bit [i] would yield OA(2100, 106, S2, 53), but
- the (dual) Plotkin bound shows that M ≥ 40 564819 207303 340847 894502 572032 / 27 > 2100 [i]
- adding a parity check bit [i] would yield OA(2100, 106, S2, 53), but
- extracting embedded orthogonal array [i] would yield OA(299, 105, S2, 52), but