MinT - Best Known (40−21, 40, s)-Nets in Base 128

Best Known (40−21, 40, s)-Nets in Base 128

(40−21, 40, 429)-Net over F₁₂₈ — Constructive and digital

Digital (19, 40, 429)-net over F₁₂₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 7, 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 (1, 11, 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]
3. digital (1, 22, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(40−21, 40, 518)-Net in Base 128 — Constructive

(19, 40, 518)-net in base 128, using

base change [i] based on digital (14, 35, 518)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (2, 12, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
  2. digital (2, 23, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(40−21, 40, 1081)-Net over F₁₂₈ — Digital

Digital (19, 40, 1081)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(40−21, 40, 5892385)-Net in Base 128 — Upper bound on s

There is no (19, 40, 5892386)-net in base 128, because

1 times m-reduction [i] would yield (19, 39, 5892386)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 15177 122560 796841 112115 628325 690341 766152 085932 876791 083484 537293 497746 608877 870352 > 128³⁹ [i]