Best Known (199−104, 199, s)-Nets in Base 2
(199−104, 199, 54)-Net over F2 — Constructive and digital
Digital (95, 199, 54)-net over F2, using
- net from sequence [i] based on digital (95, 53)-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 5 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]
(199−104, 199, 65)-Net over F2 — Digital
Digital (95, 199, 65)-net over F2, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 95 and N(F) ≥ 65, using
(199−104, 199, 199)-Net over F2 — Upper bound on s (digital)
There is no digital (95, 199, 200)-net over F2, because
- 8 times m-reduction [i] would yield digital (95, 191, 200)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2191, 200, F2, 96) (dual of [200, 9, 97]-code), but
- residual code [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
- residual code [i] would yield linear OA(295, 103, F2, 48) (dual of [103, 8, 49]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2191, 200, F2, 96) (dual of [200, 9, 97]-code), but
(199−104, 199, 202)-Net in Base 2 — Upper bound on s
There is no (95, 199, 203)-net in base 2, because
- 2 times m-reduction [i] would yield (95, 197, 203)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2197, 203, S2, 102), but
- adding a parity check bit [i] would yield OA(2198, 204, S2, 103), but
- the (dual) Plotkin bound shows that M ≥ 6 427752 177035 961102 167848 369364 650410 088811 975131 171341 205504 / 13 > 2198 [i]
- adding a parity check bit [i] would yield OA(2198, 204, S2, 103), but
- extracting embedded orthogonal array [i] would yield OA(2197, 203, S2, 102), but