MinT - Best Known (45, 87, s)-Nets in Base 3

Best Known (45, 87, s)-Nets in Base 3

(45, 87, 48)-Net over F₃ — Constructive and digital

Digital (45, 87, 48)-net over F₃, using

net from sequence [i] based on digital (45, 47)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
  - an explicitly constructive algebraic function field [i]

(45, 87, 57)-Net over F₃ — Digital

Digital (45, 87, 57)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁸⁷, 57, F₃, 5, 42) (dual of [(57, 5), 198, 43]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3⁸², 56, F₃, 5, 42) (dual of [(56, 5), 198, 43]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(5;F,237P) [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(45, 87, 391)-Net in Base 3 — Upper bound on s

There is no (45, 87, 392)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 337286 240605 917635 005694 659315 253984 787345 > 3⁸⁷ [i]