Best Known (182−85, 182, s)-Nets in Base 2
(182−85, 182, 54)-Net over F2 — Constructive and digital
Digital (97, 182, 54)-net over F2, using
- t-expansion [i] based on digital (95, 182, 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
(182−85, 182, 65)-Net over F2 — Digital
Digital (97, 182, 65)-net over F2, using
- t-expansion [i] based on digital (95, 182, 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
(182−85, 182, 250)-Net over F2 — Upper bound on s (digital)
There is no digital (97, 182, 251)-net over F2, because
- 1 times m-reduction [i] would yield digital (97, 181, 251)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2181, 251, F2, 84) (dual of [251, 70, 85]-code), but
- construction Y1 [i] would yield
- linear OA(2180, 225, F2, 84) (dual of [225, 45, 85]-code), but
- construction Y1 [i] would yield
- linear OA(2179, 209, F2, 84) (dual of [209, 30, 85]-code), but
- adding a parity check bit [i] would yield linear OA(2180, 210, F2, 85) (dual of [210, 30, 86]-code), but
- OA(245, 225, S2, 16), but
- discarding factors would yield OA(245, 189, S2, 16), but
- the Rao or (dual) Hamming bound shows that M ≥ 36 338770 420561 > 245 [i]
- discarding factors would yield OA(245, 189, S2, 16), but
- linear OA(2179, 209, F2, 84) (dual of [209, 30, 85]-code), but
- construction Y1 [i] would yield
- OA(270, 251, S2, 26), but
- discarding factors would yield OA(270, 242, S2, 26), but
- the Rao or (dual) Hamming bound shows that M ≥ 1196 766186 413200 284168 > 270 [i]
- discarding factors would yield OA(270, 242, S2, 26), but
- linear OA(2180, 225, F2, 84) (dual of [225, 45, 85]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2181, 251, F2, 84) (dual of [251, 70, 85]-code), but
(182−85, 182, 269)-Net in Base 2 — Upper bound on s
There is no (97, 182, 270)-net in base 2, because
- 1 times m-reduction [i] would yield (97, 181, 270)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 122188 760695 599057 876007 845850 679248 929938 730200 554256 > 2181 [i]