MinT - Best Known (78−50, 78, s)-Nets in Base 128

Best Known (78−50, 78, s)-Nets in Base 128

(78−50, 78, 321)-Net over F₁₂₈ — Constructive and digital

Digital (28, 78, 321)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 25, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (3, 53, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(78−50, 78, 577)-Net over F₁₂₈ — Digital

Digital (28, 78, 577)-net over F₁₂₈, using

net from sequence [i] based on digital (28, 576)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 28 and N(F) ≥ 577, using
  - a curve from the manYPoints database [i]

(78−50, 78, 300815)-Net in Base 128 — Upper bound on s

There is no (28, 78, 300816)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 230 362008 244553 610696 967620 728778 274302 847021 385730 058565 779276 038568 496101 977310 486447 886222 971383 290090 572154 165686 248518 412754 513382 505511 200282 852869 455922 065552 > 128⁷⁸ [i]