MinT - Best Known (48, 48+47, s)-Nets in Base 3

Best Known (48, 48+47, s)-Nets in Base 3

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

Digital (48, 95, 48)-net over F₃, using

t-expansion [i] based on digital (45, 95, 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]

(48, 48+47, 57)-Net over F₃ — Digital

Digital (48, 95, 57)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁹⁵, 57, F₃, 5, 47) (dual of [(57, 5), 190, 48]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3⁹⁵, 58, F₃, 5, 47) (dual of [(58, 5), 195, 48]-NRT-code), using
  - strength reduction [i] based on linear OOA(3⁹⁵, 58, F₃, 5, 48) (dual of [(58, 5), 195, 49]-NRT-code), using
    - constructionÂ X applied to AG(5;F,226P) âŠ‚ AG(5;F,234P) [i] based on
      1. linear OOA(3⁸⁸, 55, F₃, 5, 48) (dual of [(55, 5), 187, 49]-NRT-code), using algebraic-geometric NRT-code AG(5;F,226P) [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(3⁸⁰, 55, F₃, 5, 40) (dual of [(55, 5), 195, 41]-NRT-code), using algebraic-geometric NRT-code AG(5;F,234P) [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]

(48, 48+47, 398)-Net in Base 3 — Upper bound on s

There is no (48, 95, 399)-net in base 3, because

1 times m-reduction [i] would yield (48, 94, 399)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 741 228505 962149 109879 777006 399918 680789 255955 > 3⁹⁴ [i]