Best Known (175−83, 175, s)-Nets in Base 2
(175−83, 175, 53)-Net over F2 — Constructive and digital
Digital (92, 175, 53)-net over F2, using
- t-expansion [i] based on digital (90, 175, 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
(175−83, 175, 60)-Net over F2 — Digital
Digital (92, 175, 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
(175−83, 175, 227)-Net over F2 — Upper bound on s (digital)
There is no digital (92, 175, 228)-net over F2, because
- 1 times m-reduction [i] would yield digital (92, 174, 228)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2174, 228, F2, 82) (dual of [228, 54, 83]-code), but
- construction Y1 [i] would yield
- linear OA(2173, 208, F2, 82) (dual of [208, 35, 83]-code), but
- construction Y1 [i] would yield
- linear OA(2172, 196, F2, 82) (dual of [196, 24, 83]-code), but
- adding a parity check bit [i] would yield linear OA(2173, 197, F2, 83) (dual of [197, 24, 84]-code), but
- OA(235, 208, S2, 12), but
- discarding factors would yield OA(235, 173, S2, 12), but
- the Rao or (dual) Hamming bound shows that M ≥ 35365 229344 > 235 [i]
- discarding factors would yield OA(235, 173, S2, 12), but
- linear OA(2172, 196, F2, 82) (dual of [196, 24, 83]-code), but
- construction Y1 [i] would yield
- OA(254, 228, S2, 20), but
- discarding factors would yield OA(254, 195, S2, 20), but
- the Rao or (dual) Hamming bound shows that M ≥ 18304 094847 646336 > 254 [i]
- discarding factors would yield OA(254, 195, S2, 20), but
- linear OA(2173, 208, F2, 82) (dual of [208, 35, 83]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2174, 228, F2, 82) (dual of [228, 54, 83]-code), but
(175−83, 175, 249)-Net in Base 2 — Upper bound on s
There is no (92, 175, 250)-net in base 2, because
- 1 times m-reduction [i] would yield (92, 174, 250)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 24496 978873 222604 275882 899338 038718 038274 237092 551961 > 2174 [i]