MinT - Best Known (59−35, 59, s)-Nets in Base 128

Best Known (59−35, 59, s)-Nets in Base 128

(59−35, 59, 384)-Net over F₁₂₈ — Constructive and digital

Digital (24, 59, 384)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (24, 60, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 21, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]
  2. digital (3, 39, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(59−35, 59, 407)-Net in Base 128 — Constructive

(24, 59, 407)-net in base 128, using

128¹ times duplication [i] based on (23, 58, 407)-net in base 128, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 18, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. (5, 40, 257)-net in base 128, using
    - base change [i] based on digital (0, 35, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(59−35, 59, 513)-Net over F₁₂₈ — Digital

Digital (24, 59, 513)-net over F₁₂₈, using

net from sequence [i] based on digital (24, 512)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 24 and N(F) ≥ 513, using
  - a curve from the manYPoints database [i]

(59−35, 59, 873860)-Net in Base 128 — Upper bound on s

There is no (24, 59, 873861)-net in base 128, because

1 times m-reduction [i] would yield (24, 58, 873861)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 165 266008 267600 098579 577143 743466 165988 568317 464986 730504 557165 378564 320269 995044 965348 511907 173172 634425 911348 044001 435744 > 128⁵⁸ [i]