Best Known (178−89, 178, s)-Nets in Base 2
(178−89, 178, 52)-Net over F2 — Constructive and digital
Digital (89, 178, 52)-net over F2, using
- t-expansion [i] based on digital (85, 178, 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
(178−89, 178, 57)-Net over F2 — Digital
Digital (89, 178, 57)-net over F2, using
- t-expansion [i] based on digital (83, 178, 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
(178−89, 178, 190)-Net over F2 — Upper bound on s (digital)
There is no digital (89, 178, 191)-net over F2, because
- 1 times m-reduction [i] would yield digital (89, 177, 191)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2177, 191, F2, 88) (dual of [191, 14, 89]-code), but
- residual code [i] would yield linear OA(289, 102, F2, 44) (dual of [102, 13, 45]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2177, 191, F2, 88) (dual of [191, 14, 89]-code), but
(178−89, 178, 221)-Net in Base 2 — Upper bound on s
There is no (89, 178, 222)-net in base 2, because
- 1 times m-reduction [i] would yield (89, 177, 222)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 218356 515802 010523 273612 317818 156922 577578 083235 652130 > 2177 [i]