MinT - Best Known (52−43, 52, s)-Nets in Base 128

Best Known (52−43, 52, s)-Nets in Base 128

(52−43, 52, 288)-Net over F₁₂₈ — Constructive and digital

Digital (9, 52, 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]

(52−43, 52, 321)-Net in Base 128

(9, 52, 321)-net in base 128, using

4 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]

(52−43, 52, 8947)-Net in Base 128 — Upper bound on s

There is no (9, 52, 8948)-net in base 128, because

1 times m-reduction [i] would yield (9, 51, 8948)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 293854 619535 445650 736655 421342 145597 307208 281170 317295 491506 934606 952438 709590 564806 385142 007513 866966 154472 > 128⁵¹ [i]