MinT - Best Known (20−13, 20, s)-Nets in Base 128

Best Known (20−13, 20, s)-Nets in Base 128

(20−13, 20, 279)-Net over F₁₂₈ — Constructive and digital

Digital (7, 20, 279)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 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, 14, 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]

(20−13, 20, 321)-Net in Base 128

(7, 20, 321)-net in base 128, using

20 times m-reduction [i] based on (7, 40, 321)-net in base 128, using
- base change [i] based on digital (2, 35, 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]

(20−13, 20, 110978)-Net in Base 128 — Upper bound on s

There is no (7, 20, 110979)-net in base 128, because

1 times m-reduction [i] would yield (7, 19, 110979)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 10889 162535 061036 350728 995345 469515 465312 > 128¹⁹ [i]