MinT - Best Known (16−14, 16, s)-Nets in Base 128

Best Known (16−14, 16, s)-Nets in Base 128

(16−14, 16, 150)-Net over F₁₂₈ — Constructive and digital

Digital (2, 16, 150)-net over F₁₂₈, using

t-expansion [i] based on digital (1, 16, 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]

(16−14, 16, 172)-Net over F₁₂₈ — Digital

Digital (2, 16, 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]

(16−14, 16, 257)-Net in Base 128 — Constructive

(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]

(16−14, 16, 1741)-Net in Base 128 — Upper bound on s

There is no (2, 16, 1742)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 5210 235927 731763 069500 605168 151724 > 128¹⁶ [i]