Best Known (199−105, 199, s)-Nets in Base 2
(199−105, 199, 53)-Net over F2 — Constructive and digital
Digital (94, 199, 53)-net over F2, using
- t-expansion [i] based on 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]
- net from sequence [i] based on digital (90, 52)-sequence over F2, using
(199−105, 199, 60)-Net over F2 — Digital
Digital (94, 199, 60)-net over F2, using
- t-expansion [i] based on digital (92, 199, 60)-net over F2, using
- net from sequence [i] based on digital (92, 59)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 92 and N(F) ≥ 60, using
- net from sequence [i] based on digital (92, 59)-sequence over F2, using
(199−105, 199, 198)-Net over F2 — Upper bound on s (digital)
There is no digital (94, 199, 199)-net over F2, because
- 9 times m-reduction [i] would yield digital (94, 190, 199)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2190, 199, F2, 96) (dual of [199, 9, 97]-code), but
- residual code [i] would yield linear OA(294, 102, F2, 48) (dual of [102, 8, 49]-code), but
- residual code [i] would yield linear OA(246, 53, F2, 24) (dual of [53, 7, 25]-code), but
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
- residual code [i] would yield linear OA(246, 53, F2, 24) (dual of [53, 7, 25]-code), but
- residual code [i] would yield linear OA(294, 102, F2, 48) (dual of [102, 8, 49]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2190, 199, F2, 96) (dual of [199, 9, 97]-code), but
(199−105, 199, 200)-Net in Base 2 — Upper bound on s
There is no (94, 199, 201)-net in base 2, because
- 5 times m-reduction [i] would yield (94, 194, 201)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2194, 201, S2, 100), but
- adding a parity check bit [i] would yield OA(2195, 202, S2, 101), but
- the (dual) Plotkin bound shows that M ≥ 3 213876 088517 980551 083924 184682 325205 044405 987565 585670 602752 / 51 > 2195 [i]
- adding a parity check bit [i] would yield OA(2195, 202, S2, 101), but
- extracting embedded orthogonal array [i] would yield OA(2194, 201, S2, 100), but