MinT - Best Known (50−42, 50, s)-Nets in Base 128

Best Known (50−42, 50, s)-Nets in Base 128

(50−42, 50, 216)-Net over F₁₂₈ — Constructive and digital

Digital (8, 50, 216)-net over F₁₂₈, using

t-expansion [i] based on digital (5, 50, 216)-net over F₁₂₈, using
- net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
    - a function field by SÃ©mirat [i]

(50−42, 50, 258)-Net in Base 128 — Constructive

(8, 50, 258)-net in base 128, using

6 times m-reduction [i] based on (8, 56, 258)-net in base 128, using
- base change [i] based on digital (1, 49, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(50−42, 50, 276)-Net over F₁₂₈ — Digital

Digital (8, 50, 276)-net over F₁₂₈, using

net from sequence [i] based on digital (8, 275)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 8 and N(F) ≥ 276, using
  - a curve from the manYPoints database [i]

(50−42, 50, 289)-Net in Base 128

(8, 50, 289)-net in base 128, using

6 times m-reduction [i] based on (8, 56, 289)-net in base 128, using
- base change [i] based on digital (1, 49, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(50−42, 50, 7099)-Net in Base 128 — Upper bound on s

There is no (8, 50, 7100)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 2296 171583 031044 858993 674485 452287 967642 747120 652283 732101 874287 510084 154688 613505 716561 665103 852661 255186 > 128⁵⁰ [i]