MinT - Best Known (51−49, 51, s)-Nets in Base 128

Best Known (51−49, 51, s)-Nets in Base 128

(51−49, 51, 150)-Net over F₁₂₈ — Constructive and digital

Digital (2, 51, 150)-net over F₁₂₈, using

t-expansion [i] based on digital (1, 51, 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]

(51−49, 51, 172)-Net over F₁₂₈ — Digital

Digital (2, 51, 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]

(51−49, 51, 1525)-Net in Base 128 — Upper bound on s

There is no (2, 51, 1526)-net in base 128, because

27 times m-reduction [i] would yield (2, 24, 1526)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 376 418960 441108 647243 139541 425020 807708 678656 164172 > 128²⁴ [i]