Best Known (169−85, 169, s)-Nets in Base 2
(169−85, 169, 51)-Net over F2 — Constructive and digital
Digital (84, 169, 51)-net over F2, using
- t-expansion [i] based on digital (80, 169, 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
(169−85, 169, 57)-Net over F2 — Digital
Digital (84, 169, 57)-net over F2, using
- t-expansion [i] based on digital (83, 169, 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
(169−85, 169, 179)-Net over F2 — Upper bound on s (digital)
There is no digital (84, 169, 180)-net over F2, because
- 1 times m-reduction [i] would yield digital (84, 168, 180)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2168, 180, F2, 84) (dual of [180, 12, 85]-code), but
- residual code [i] would yield linear OA(284, 95, F2, 42) (dual of [95, 11, 43]-code), but
- adding a parity check bit [i] would yield linear OA(285, 96, F2, 43) (dual of [96, 11, 44]-code), but
- “Bro†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(285, 96, F2, 43) (dual of [96, 11, 44]-code), but
- residual code [i] would yield linear OA(284, 95, F2, 42) (dual of [95, 11, 43]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2168, 180, F2, 84) (dual of [180, 12, 85]-code), but
(169−85, 169, 207)-Net in Base 2 — Upper bound on s
There is no (84, 169, 208)-net in base 2, because
- 1 times m-reduction [i] would yield (84, 168, 208)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 393 520451 869462 700436 624567 826565 744117 014287 558510 > 2168 [i]