MinT - Best Known (8, 8+37, s)-Nets in Base 128

Best Known (8, 8+37, s)-Nets in Base 128

(8, 8+37, 216)-Net over F₁₂₈ — Constructive and digital

Digital (8, 45, 216)-net over F₁₂₈, using

t-expansion [i] based on digital (5, 45, 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]

(8, 8+37, 259)-Net in Base 128 — Constructive

(8, 45, 259)-net in base 128, using

3 times m-reduction [i] based on (8, 48, 259)-net in base 128, using
- base change [i] based on digital (2, 42, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(8, 8+37, 276)-Net over F₁₂₈ — Digital

Digital (8, 45, 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]

(8, 8+37, 321)-Net in Base 128

(8, 45, 321)-net in base 128, using

3 times m-reduction [i] based on (8, 48, 321)-net in base 128, using
- base change [i] based on digital (2, 42, 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]

(8, 8+37, 8410)-Net in Base 128 — Upper bound on s

There is no (8, 45, 8411)-net in base 128, because

1 times m-reduction [i] would yield (8, 44, 8411)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 521 686208 705121 155931 200453 326605 773527 456900 028368 679908 603859 271210 047648 934760 661088 007007 > 128⁴⁴ [i]