MinT - Best Known (8, 27, s)-Nets in Base 128

Best Known (8, 27, s)-Nets in Base 128

(8, 27, 216)-Net over F₁₂₈ — Constructive and digital

Digital (8, 27, 216)-net over F₁₂₈, using

t-expansion [i] based on digital (5, 27, 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, 27, 261)-Net in Base 128 — Constructive

(8, 27, 261)-net in base 128, using

5 times m-reduction [i] based on (8, 32, 261)-net in base 128, using
- base change [i] based on digital (4, 28, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(8, 27, 276)-Net over F₁₂₈ — Digital

Digital (8, 27, 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, 27, 321)-Net in Base 128

(8, 27, 321)-net in base 128, using

21 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, 27, 39939)-Net in Base 128 — Upper bound on s

There is no (8, 27, 39940)-net in base 128, because

1 times m-reduction [i] would yield (8, 26, 39940)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 6 131141 774997 860826 995295 848452 988372 929185 108860 243072 > 128²⁶ [i]