MinT - Best Known (42−31, 42, s)-Nets in Base 128

Best Known (42−31, 42, s)-Nets in Base 128

(42−31, 42, 288)-Net over F₁₂₈ — Constructive and digital

Digital (11, 42, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 42, 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]

(42−31, 42, 296)-Net over F₁₂₈ — Digital

Digital (11, 42, 296)-net over F₁₂₈, using

t-expansion [i] based on digital (10, 42, 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]

(42−31, 42, 321)-Net in Base 128

(11, 42, 321)-net in base 128, using

30 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]

(42−31, 42, 29077)-Net in Base 128 — Upper bound on s

There is no (11, 42, 29078)-net in base 128, because

1 times m-reduction [i] would yield (11, 41, 29078)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 248 667255 446974 164264 270026 576706 953734 033632 395126 089950 295257 139696 888542 694410 250312 > 128⁴¹ [i]