MinT - Best Known (10−8, 10, s)-Nets in Base 128

Best Known (10−8, 10, s)-Nets in Base 128

(10−8, 10, 150)-Net over F₁₂₈ — Constructive and digital

Digital (2, 10, 150)-net over F₁₂₈, using

t-expansion [i] based on digital (1, 10, 150)-net over F₁₂₈, using
- net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
    - a function field by SÃ©mirat [i]

(10−8, 10, 172)-Net over F₁₂₈ — Digital

Digital (2, 10, 172)-net over F₁₂₈, using

net from sequence [i] based on digital (2, 171)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 2 and N(F) ≥ 172, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(10−8, 10, 257)-Net in Base 128 — Constructive

(2, 10, 257)-net in base 128, using

6 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
- base change [i] based on digital (0, 14, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]

(10−8, 10, 3229)-Net in Base 128 — Upper bound on s

There is no (2, 10, 3230)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1182 053416 876909 825191 > 128¹⁰ [i]