MinT - Best Known (31−21, 31, s)-Nets in Base 128

Best Known (31−21, 31, s)-Nets in Base 128

(31−21, 31, 288)-Net over F₁₂₈ — Constructive and digital

Digital (10, 31, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 31, 288)-net over F₁₂₈, using
- net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
    - a function field by SÃ©mirat [i]

(31−21, 31, 296)-Net over F₁₂₈ — Digital

Digital (10, 31, 296)-net over F₁₂₈, using

net from sequence [i] based on digital (10, 295)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 10 and N(F) ≥ 296, using
  - a curve from the manYPoints database [i]

(31−21, 31, 321)-Net in Base 128

(10, 31, 321)-net in base 128, using

33 times m-reduction [i] based on (10, 64, 321)-net in base 128, using
- base change [i] based on digital (2, 56, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(31−21, 31, 74778)-Net in Base 128 — Upper bound on s

There is no (10, 31, 74779)-net in base 128, because

1 times m-reduction [i] would yield (10, 30, 74779)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1645 646749 743830 551960 232003 016990 163196 887965 912753 337112 677239 > 128³⁰ [i]