Best Known (164−78, 164, s)-Nets in Base 2
(164−78, 164, 52)-Net over F2 — Constructive and digital
Digital (86, 164, 52)-net over F2, using
- t-expansion [i] based on digital (85, 164, 52)-net over F2, using
- net from sequence [i] based on digital (85, 51)-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 3 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 (85, 51)-sequence over F2, using
(164−78, 164, 57)-Net over F2 — Digital
Digital (86, 164, 57)-net over F2, using
- t-expansion [i] based on digital (83, 164, 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
(164−78, 164, 192)-Net over F2 — Upper bound on s (digital)
There is no digital (86, 164, 193)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2164, 193, F2, 78) (dual of [193, 29, 79]-code), but
- 2 times code embedding in larger space [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
- 2 times code embedding in larger space [i] would yield linear OA(2166, 195, F2, 78) (dual of [195, 29, 79]-code), but
(164−78, 164, 230)-Net in Base 2 — Upper bound on s
There is no (86, 164, 231)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 23 844219 636832 585656 226689 780016 767472 709797 533978 > 2164 [i]