Best Known (50, 105, s)-Nets in Base 2
(50, 105, 35)-Net over F2 — Constructive and digital
Digital (50, 105, 35)-net over F2, using
- t-expansion [i] based on digital (48, 105, 35)-net over F2, using
- net from sequence [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]
- net from sequence [i] based on digital (48, 34)-sequence over F2, using
(50, 105, 40)-Net over F2 — Digital
Digital (50, 105, 40)-net over F2, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 50 and N(F) ≥ 40, using
(50, 105, 109)-Net over F2 — Upper bound on s (digital)
There is no digital (50, 105, 110)-net over F2, because
- 3 times m-reduction [i] would yield digital (50, 102, 110)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2102, 110, F2, 52) (dual of [110, 8, 53]-code), but
- residual code [i] would yield linear OA(250, 57, F2, 26) (dual of [57, 7, 27]-code), but
- adding a parity check bit [i] would yield linear OA(251, 58, F2, 27) (dual of [58, 7, 28]-code), but
- “vT3†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(251, 58, F2, 27) (dual of [58, 7, 28]-code), but
- residual code [i] would yield linear OA(250, 57, F2, 26) (dual of [57, 7, 27]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2102, 110, F2, 52) (dual of [110, 8, 53]-code), but
(50, 105, 111)-Net in Base 2 — Upper bound on s
There is no (50, 105, 112)-net in base 2, because
- 5 times m-reduction [i] would yield (50, 100, 112)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2100, 112, S2, 50), but
- the linear programming bound shows that M ≥ 552 695661 699508 019052 562597 543936 / 403 > 2100 [i]
- extracting embedded orthogonal array [i] would yield OA(2100, 112, S2, 50), but