MinT - Best Known (78−48, 78, s)-Nets in Base 128

Best Known (78−48, 78, s)-Nets in Base 128

(78−48, 78, 384)-Net over F₁₂₈ — Constructive and digital

Digital (30, 78, 384)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 27, 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]
2. digital (3, 51, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(78−48, 78, 609)-Net over F₁₂₈ — Digital

Digital (30, 78, 609)-net over F₁₂₈, using

net from sequence [i] based on digital (30, 608)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 30 and N(F) ≥ 609, using
  - a curve from the manYPoints database [i]

(78−48, 78, 544480)-Net in Base 128 — Upper bound on s

There is no (30, 78, 544481)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 230 352483 127620 047665 216678 702869 076313 336631 366882 791930 789100 544966 614994 758666 262557 627615 960880 009656 122033 472420 868746 963753 391633 407477 629641 971838 792608 989332 > 128⁷⁸ [i]