Best Known (89, 89+91, s)-Nets in Base 2
(89, 89+91, 52)-Net over F2 — Constructive and digital
Digital (89, 180, 52)-net over F2, using
- t-expansion [i] based on digital (85, 180, 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
(89, 89+91, 57)-Net over F2 — Digital
Digital (89, 180, 57)-net over F2, using
- t-expansion [i] based on digital (83, 180, 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
(89, 89+91, 189)-Net over F2 — Upper bound on s (digital)
There is no digital (89, 180, 190)-net over F2, because
- 1 times m-reduction [i] would yield digital (89, 179, 190)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2179, 190, F2, 90) (dual of [190, 11, 91]-code), but
- residual code [i] would yield linear OA(289, 99, F2, 45) (dual of [99, 10, 46]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2179, 190, F2, 90) (dual of [190, 11, 91]-code), but
(89, 89+91, 217)-Net in Base 2 — Upper bound on s
There is no (89, 180, 218)-net in base 2, because
- 1 times m-reduction [i] would yield (89, 179, 218)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 866037 660008 562550 987667 976936 051628 110575 734968 835720 > 2179 [i]