MinT - Best Known (78−49, 78, s)-Nets in Base 128

Best Known (78−49, 78, s)-Nets in Base 128

(78−49, 78, 345)-Net over F₁₂₈ — Constructive and digital

Digital (29, 78, 345)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 24, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (5, 54, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(78−49, 78, 577)-Net over F₁₂₈ — Digital

Digital (29, 78, 577)-net over F₁₂₈, using

t-expansion [i] based on digital (28, 78, 577)-net over F₁₂₈, using
- net from sequence [i] based on digital (28, 576)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 28 and N(F) ≥ 577, using
    - a curve from the manYPoints database [i]

(78−49, 78, 444815)-Net in Base 128 — Upper bound on s

There is no (29, 78, 444816)-net in base 128, because

1 times m-reduction [i] would yield (29, 77, 444816)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1 799646 213895 986294 923972 705220 801917 622135 697261 645691 170940 054077 809911 393911 307702 756117 576859 069703 968169 608766 227707 702784 539497 983332 406058 902539 288895 744488 > 128⁷⁷ [i]