Best Known (93, 93+106, s)-Nets in Base 2
(93, 93+106, 53)-Net over F2 — Constructive and digital
Digital (93, 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
(93, 93+106, 60)-Net over F2 — Digital
Digital (93, 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
(93, 93+106, 195)-Net over F2 — Upper bound on s (digital)
There is no digital (93, 199, 196)-net over F2, because
- 10 times m-reduction [i] would yield digital (93, 189, 196)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2189, 196, F2, 96) (dual of [196, 7, 97]-code), but
(93, 93+106, 198)-Net in Base 2 — Upper bound on s
There is no (93, 199, 199)-net in base 2, because
- 6 times m-reduction [i] would yield (93, 193, 199)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2193, 199, S2, 100), but
- adding a parity check bit [i] would yield OA(2194, 200, S2, 101), but
- the (dual) Plotkin bound shows that M ≥ 1 606938 044258 990275 541962 092341 162602 522202 993782 792835 301376 / 51 > 2194 [i]
- adding a parity check bit [i] would yield OA(2194, 200, S2, 101), but
- extracting embedded orthogonal array [i] would yield OA(2193, 199, S2, 100), but