MinT - Best Known (48−39, 48, s)-Nets in Base 128

Best Known (48−39, 48, s)-Nets in Base 128

(48−39, 48, 288)-Net over F₁₂₈ — Constructive and digital

Digital (9, 48, 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]

(48−39, 48, 321)-Net in Base 128

(9, 48, 321)-net in base 128, using

8 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]

(48−39, 48, 10176)-Net in Base 128 — Upper bound on s

There is no (9, 48, 10177)-net in base 128, because

1 times m-reduction [i] would yield (9, 47, 10177)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1094 878014 497034 668676 950080 928688 425927 005388 409038 508891 346243 591417 502196 295432 118595 366616 547264 > 128⁴⁷ [i]