Best Known (161−77, 161, s)-Nets in Base 2
(161−77, 161, 51)-Net over F2 — Constructive and digital
Digital (84, 161, 51)-net over F2, using
- t-expansion [i] based on digital (80, 161, 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
(161−77, 161, 57)-Net over F2 — Digital
Digital (84, 161, 57)-net over F2, using
- t-expansion [i] based on digital (83, 161, 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
(161−77, 161, 192)-Net over F2 — Upper bound on s (digital)
There is no digital (84, 161, 193)-net over F2, because
- 1 times m-reduction [i] would yield digital (84, 160, 193)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2160, 193, F2, 76) (dual of [193, 33, 77]-code), but
- 3 times code embedding in larger space [i] would yield linear OA(2163, 196, F2, 76) (dual of [196, 33, 77]-code), but
- adding a parity check bit [i] would yield linear OA(2164, 197, F2, 77) (dual of [197, 33, 78]-code), but
- 3 times code embedding in larger space [i] would yield linear OA(2163, 196, F2, 76) (dual of [196, 33, 77]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2160, 193, F2, 76) (dual of [193, 33, 77]-code), but
(161−77, 161, 226)-Net in Base 2 — Upper bound on s
There is no (84, 161, 227)-net in base 2, because
- 1 times m-reduction [i] would yield (84, 160, 227)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 590955 737048 800144 541130 001689 197637 637255 017440 > 2160 [i]