Best Known (86, 86+76, s)-Nets in Base 2
(86, 86+76, 52)-Net over F2 — Constructive and digital
Digital (86, 162, 52)-net over F2, using
- t-expansion [i] based on digital (85, 162, 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
(86, 86+76, 57)-Net over F2 — Digital
Digital (86, 162, 57)-net over F2, using
- t-expansion [i] based on 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
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
(86, 86+76, 194)-Net over F2 — Upper bound on s (digital)
There is no digital (86, 162, 195)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2162, 195, F2, 76) (dual of [195, 33, 77]-code), but
- 1 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
- 1 times code embedding in larger space [i] would yield linear OA(2163, 196, F2, 76) (dual of [196, 33, 77]-code), but
(86, 86+76, 236)-Net in Base 2 — Upper bound on s
There is no (86, 162, 237)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 6 192073 595537 775009 973820 517045 495606 534092 348800 > 2162 [i]