Best Known (83, 83+79, s)-Nets in Base 2
(83, 83+79, 51)-Net over F2 — Constructive and digital
Digital (83, 162, 51)-net over F2, using
- t-expansion [i] based on digital (80, 162, 51)-net over F2, using
- net from sequence [i] based on digital (80, 50)-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 2 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 (80, 50)-sequence over F2, using
(83, 83+79, 57)-Net over F2 — Digital
Digital (83, 162, 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
(83, 83+79, 189)-Net over F2 — Upper bound on s (digital)
There is no digital (83, 162, 190)-net over F2, because
- 1 times m-reduction [i] would yield digital (83, 161, 190)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2161, 190, F2, 78) (dual of [190, 29, 79]-code), but
- 5 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
- 5 times code embedding in larger space [i] would yield linear OA(2166, 195, F2, 78) (dual of [195, 29, 79]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2161, 190, F2, 78) (dual of [190, 29, 79]-code), but
(83, 83+79, 216)-Net in Base 2 — Upper bound on s
There is no (83, 162, 217)-net in base 2, because
- 1 times m-reduction [i] would yield (83, 161, 217)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 214147 424615 669477 323681 519148 870518 842439 106560 > 2161 [i]