Best Known (199−109, 199, s)-Nets in Base 2
(199−109, 199, 53)-Net over F2 — Constructive and digital
Digital (90, 199, 53)-net over F2, using
- net from sequence [i] based on digital (90, 52)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 4 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
(199−109, 199, 57)-Net over F2 — Digital
Digital (90, 199, 57)-net over F2, using
- t-expansion [i] based on digital (83, 199, 57)-net over F2, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
(199−109, 199, 191)-Net over F2 — Upper bound on s (digital)
There is no digital (90, 199, 192)-net over F2, because
- 17 times m-reduction [i] would yield digital (90, 182, 192)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2182, 192, F2, 92) (dual of [192, 10, 93]-code), but
- residual code [i] would yield linear OA(290, 99, F2, 46) (dual of [99, 9, 47]-code), but
- adding a parity check bit [i] would yield linear OA(291, 100, F2, 47) (dual of [100, 9, 48]-code), but
- “Bro†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(291, 100, F2, 47) (dual of [100, 9, 48]-code), but
- residual code [i] would yield linear OA(290, 99, F2, 46) (dual of [99, 9, 47]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2182, 192, F2, 92) (dual of [192, 10, 93]-code), but
(199−109, 199, 192)-Net in Base 2 — Upper bound on s
There is no (90, 199, 193)-net in base 2, because
- 13 times m-reduction [i] would yield (90, 186, 193)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2186, 193, S2, 96), but
- adding a parity check bit [i] would yield OA(2187, 194, S2, 97), but
- the (dual) Plotkin bound shows that M ≥ 12554 203470 773361 527671 578846 415332 832204 710888 928069 025792 / 49 > 2187 [i]
- adding a parity check bit [i] would yield OA(2187, 194, S2, 97), but
- extracting embedded orthogonal array [i] would yield OA(2186, 193, S2, 96), but