MinT - Best Known (51−40, 51, s)-Nets in Base 128

Best Known (51−40, 51, s)-Nets in Base 128

(51−40, 51, 288)-Net over F₁₂₈ — Constructive and digital

Digital (11, 51, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 51, 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]

(51−40, 51, 296)-Net over F₁₂₈ — Digital

Digital (11, 51, 296)-net over F₁₂₈, using

t-expansion [i] based on digital (10, 51, 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]

(51−40, 51, 321)-Net in Base 128

(11, 51, 321)-net in base 128, using

21 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]

(51−40, 51, 15438)-Net in Base 128 — Upper bound on s

There is no (11, 51, 15439)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 293605 393261 889831 132823 623684 781271 237360 426499 504469 159901 257254 036608 215333 354410 859209 362277 034622 827844 > 128⁵¹ [i]