Best Known (170−83, 170, s)-Nets in Base 2
(170−83, 170, 52)-Net over F2 — Constructive and digital
Digital (87, 170, 52)-net over F2, using
- t-expansion [i] based on digital (85, 170, 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
(170−83, 170, 57)-Net over F2 — Digital
Digital (87, 170, 57)-net over F2, using
- t-expansion [i] based on digital (83, 170, 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
(170−83, 170, 192)-Net over F2 — Upper bound on s (digital)
There is no digital (87, 170, 193)-net over F2, because
- 1 times m-reduction [i] would yield digital (87, 169, 193)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2169, 193, F2, 82) (dual of [193, 24, 83]-code), but
- 3 times code embedding in larger space [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
- 3 times code embedding in larger space [i] would yield linear OA(2172, 196, F2, 82) (dual of [196, 24, 83]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2169, 193, F2, 82) (dual of [193, 24, 83]-code), but
(170−83, 170, 225)-Net in Base 2 — Upper bound on s
There is no (87, 170, 226)-net in base 2, because
- 1 times m-reduction [i] would yield (87, 169, 226)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 812 857539 831963 759071 442382 180236 508147 759060 132296 > 2169 [i]