MinT - Best Known (56, 56+54, s)-Nets in Base 3

Best Known (56, 56+54, s)-Nets in Base 3

(56, 56+54, 52)-Net over F₃ — Constructive and digital

Digital (56, 110, 52)-net over F₃, using

2 times m-reduction [i] based on digital (56, 112, 52)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 41, 24)-net over F₃, using
    - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
  2. digital (15, 71, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]

(56, 56+54, 65)-Net over F₃ — Digital

Digital (56, 110, 65)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹¹⁰, 65, F₃, 5, 54) (dual of [(65, 5), 215, 55]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹¹⁰, 66, F₃, 5, 54) (dual of [(66, 5), 220, 55]-NRT-code), using
  - constructionÂ X applied to AG(5;F,260P) âŠ‚ AG(5;F,268P) [i] based on
    1. linear OOA(3¹⁰³, 63, F₃, 5, 54) (dual of [(63, 5), 212, 55]-NRT-code), using algebraic-geometric NRT-code AG(5;F,260P) [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
      - a curve from the manYPoints database [i]
    2. linear OOA(3⁹⁵, 63, F₃, 5, 46) (dual of [(63, 5), 220, 47]-NRT-code), using algebraic-geometric NRT-code AG(5;F,268P) [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64 (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]

(56, 56+54, 454)-Net in Base 3 — Upper bound on s

There is no (56, 110, 455)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 32098 895889 738626 188220 824002 265405 634391 255542 917739 > 3¹¹⁰ [i]