MinT - Best Known (44−35, 44, s)-Nets in Base 128

Best Known (44−35, 44, s)-Nets in Base 128

(44−35, 44, 288)-Net over F₁₂₈ — Constructive and digital

Digital (9, 44, 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]

(44−35, 44, 321)-Net in Base 128

(9, 44, 321)-net in base 128, using

12 times m-reduction [i] based on (9, 56, 321)-net in base 128, using
- base change [i] based on digital (2, 49, 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]

(44−35, 44, 12073)-Net in Base 128 — Upper bound on s

There is no (9, 44, 12074)-net in base 128, because

1 times m-reduction [i] would yield (9, 43, 12074)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 4 074140 578376 767479 315849 405077 955679 662778 755169 006122 539933 642742 032285 859455 695883 779982 > 128⁴³ [i]