Best Known (168−78, 168, s)-Nets in Base 2
(168−78, 168, 53)-Net over F2 — Constructive and digital
Digital (90, 168, 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]
(168−78, 168, 57)-Net over F2 — Digital
Digital (90, 168, 57)-net over F2, using
- t-expansion [i] based on digital (83, 168, 57)-net over F2, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
(168−78, 168, 236)-Net over F2 — Upper bound on s (digital)
There is no digital (90, 168, 237)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2168, 237, F2, 78) (dual of [237, 69, 79]-code), but
- construction Y1 [i] would yield
- linear OA(2167, 211, F2, 78) (dual of [211, 44, 79]-code), but
- construction Y1 [i] would yield
- linear OA(2166, 195, F2, 78) (dual of [195, 29, 79]-code), but
- adding a parity check bit [i] would yield linear OA(2167, 196, F2, 79) (dual of [196, 29, 80]-code), but
- OA(244, 211, S2, 16), but
- discarding factors would yield OA(244, 173, S2, 16), but
- the Rao or (dual) Hamming bound shows that M ≥ 17 734855 699135 > 244 [i]
- discarding factors would yield OA(244, 173, S2, 16), but
- linear OA(2166, 195, F2, 78) (dual of [195, 29, 79]-code), but
- construction Y1 [i] would yield
- OA(269, 237, S2, 26), but
- discarding factors would yield OA(269, 230, S2, 26), but
- the Rao or (dual) Hamming bound shows that M ≥ 609 142116 361774 291812 > 269 [i]
- discarding factors would yield OA(269, 230, S2, 26), but
- linear OA(2167, 211, F2, 78) (dual of [211, 44, 79]-code), but
- construction Y1 [i] would yield
(168−78, 168, 251)-Net in Base 2 — Upper bound on s
There is no (90, 168, 252)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 400 201029 545177 600255 025221 619433 435902 167208 811474 > 2168 [i]