MinT - Best Known (75−49, 75, s)-Nets in Base 128

Best Known (75−49, 75, s)-Nets in Base 128

(75−49, 75, 300)-Net over F₁₂₈ — Constructive and digital

Digital (26, 75, 300)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 25, 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]
2. digital (1, 50, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(75−49, 75, 513)-Net over F₁₂₈ — Digital

Digital (26, 75, 513)-net over F₁₂₈, using

t-expansion [i] based on digital (24, 75, 513)-net over F₁₂₈, using
- net from sequence [i] based on digital (24, 512)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 24 and N(F) ≥ 513, using
    - a curve from the manYPoints database [i]

(75−49, 75, 242531)-Net in Base 128 — Upper bound on s

There is no (26, 75, 242532)-net in base 128, because

1 times m-reduction [i] would yield (26, 74, 242532)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 858110 304095 918036 716150 260072 500647 816054 305286 375136 832866 321235 013049 500571 619196 992578 130654 983389 081110 335298 032975 240149 679302 740395 666552 227030 441364 > 128⁷⁴ [i]