MinT - Best Known (47, 47+45, s)-Nets in Base 3

Best Known (47, 47+45, s)-Nets in Base 3

(47, 47+45, 48)-Net over F₃ — Constructive and digital

Digital (47, 92, 48)-net over F₃, using

t-expansion [i] based on digital (45, 92, 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]

(47, 47+45, 57)-Net over F₃ — Digital

Digital (47, 92, 57)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁹², 57, F₃, 5, 45) (dual of [(57, 5), 193, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3⁹², 58, F₃, 5, 45) (dual of [(58, 5), 198, 46]-NRT-code), using
  - constructionÂ X applied to AG(5;F,229P) âŠ‚ AG(5;F,237P) [i] based on
    1. linear OOA(3⁸⁵, 55, F₃, 5, 45) (dual of [(55, 5), 190, 46]-NRT-code), using algebraic-geometric NRT-code AG(5;F,229P) [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
      - a curve from the manYPoints database [i]
    2. linear OOA(3⁷⁷, 55, F₃, 5, 37) (dual of [(55, 5), 198, 38]-NRT-code), using algebraic-geometric NRT-code AG(5;F,237P) [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56 (see above)
    3. linear OOA(3⁷, 3, F₃, 5, 7) (dual of [(3, 5), 8, 8]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(5;8,3) [i]

(47, 47+45, 404)-Net in Base 3 — Upper bound on s

There is no (47, 92, 405)-net in base 3, because

1 times m-reduction [i] would yield (47, 91, 405)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 26 315494 933292 903784 202581 660222 847021 973281 > 3⁹¹ [i]