Best Known (90, 90+86, s)-Nets in Base 2
(90, 90+86, 53)-Net over F2 — Constructive and digital
Digital (90, 176, 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]
(90, 90+86, 57)-Net over F2 — Digital
Digital (90, 176, 57)-net over F2, using
- t-expansion [i] based on digital (83, 176, 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
(90, 90+86, 195)-Net over F2 — Upper bound on s (digital)
There is no digital (90, 176, 196)-net over F2, because
- 4 times m-reduction [i] would yield digital (90, 172, 196)-net over F2, but
- extracting embedded orthogonal array [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
- extracting embedded orthogonal array [i] would yield linear OA(2172, 196, F2, 82) (dual of [196, 24, 83]-code), but
(90, 90+86, 229)-Net in Base 2 — Upper bound on s
There is no (90, 176, 230)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 96725 411182 297479 291621 072627 322798 658699 588719 123248 > 2176 [i]