MinT - Best Known (56−45, 56, s)-Nets in Base 128

Best Known (56−45, 56, s)-Nets in Base 128

(56−45, 56, 288)-Net over F₁₂₈ — Constructive and digital

Digital (11, 56, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 56, 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]

(56−45, 56, 296)-Net over F₁₂₈ — Digital

Digital (11, 56, 296)-net over F₁₂₈, using

t-expansion [i] based on digital (10, 56, 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]

(56−45, 56, 321)-Net in Base 128

(11, 56, 321)-net in base 128, using

16 times m-reduction [i] based on (11, 72, 321)-net in base 128, using
- base change [i] based on digital (2, 63, 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]

(56−45, 56, 13204)-Net in Base 128 — Upper bound on s

There is no (11, 56, 13205)-net in base 128, because

1 times m-reduction [i] would yield (11, 55, 13205)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 78 863942 673124 333838 564414 910824 460409 800236 140876 333033 201862 389894 738412 124613 517014 165702 259107 661006 001395 207616 > 128⁵⁵ [i]