Best Known (50, 50+58, s)-Nets in Base 2
(50, 50+58, 35)-Net over F2 — Constructive and digital
Digital (50, 108, 35)-net over F2, using
- t-expansion [i] based on digital (48, 108, 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, 50+58, 40)-Net over F2 — Digital
Digital (50, 108, 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, 50+58, 109)-Net over F2 — Upper bound on s (digital)
There is no digital (50, 108, 110)-net over F2, because
- 6 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, 50+58, 110)-Net in Base 2 — Upper bound on s
There is no (50, 108, 111)-net in base 2, because
- 2 times m-reduction [i] would yield (50, 106, 111)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2106, 111, S2, 56), but
- adding a parity check bit [i] would yield OA(2107, 112, S2, 57), but
- the (dual) Plotkin bound shows that M ≥ 5192 296858 534827 628530 496329 220096 / 29 > 2107 [i]
- adding a parity check bit [i] would yield OA(2107, 112, S2, 57), but
- extracting embedded orthogonal array [i] would yield OA(2106, 111, S2, 56), but