Best Known (98, 98+86, s)-Nets in Base 2
(98, 98+86, 54)-Net over F2 — Constructive and digital
Digital (98, 184, 54)-net over F2, using
- t-expansion [i] based on digital (95, 184, 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]
- net from sequence [i] based on digital (95, 53)-sequence over F2, using
(98, 98+86, 65)-Net over F2 — Digital
Digital (98, 184, 65)-net over F2, using
- t-expansion [i] based on digital (95, 184, 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
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
(98, 98+86, 248)-Net over F2 — Upper bound on s (digital)
There is no digital (98, 184, 249)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2184, 249, F2, 86) (dual of [249, 65, 87]-code), but
- construction Y1 [i] would yield
- linear OA(2183, 225, F2, 86) (dual of [225, 42, 87]-code), but
- construction Y1 [i] would yield
- linear OA(2182, 211, F2, 86) (dual of [211, 29, 87]-code), but
- adding a parity check bit [i] would yield linear OA(2183, 212, F2, 87) (dual of [212, 29, 88]-code), but
- OA(242, 225, S2, 14), but
- discarding factors would yield OA(242, 219, S2, 14), but
- the Rao or (dual) Hamming bound shows that M ≥ 4 498367 189624 > 242 [i]
- discarding factors would yield OA(242, 219, S2, 14), but
- linear OA(2182, 211, F2, 86) (dual of [211, 29, 87]-code), but
- construction Y1 [i] would yield
- OA(265, 249, S2, 24), but
- discarding factors would yield OA(265, 231, S2, 24), but
- the Rao or (dual) Hamming bound shows that M ≥ 38 110539 835438 477924 > 265 [i]
- discarding factors would yield OA(265, 231, S2, 24), but
- linear OA(2183, 225, F2, 86) (dual of [225, 42, 87]-code), but
- construction Y1 [i] would yield
(98, 98+86, 268)-Net in Base 2 — Upper bound on s
There is no (98, 184, 269)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 24 657425 470763 762193 457759 359220 903158 156419 015792 210688 > 2184 [i]