MinT - Best Known (75−45, 75, s)-Nets in Base 128

Best Known (75−45, 75, s)-Nets in Base 128

(75−45, 75, 408)-Net over F₁₂₈ — Constructive and digital

Digital (30, 75, 408)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 25, 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 (5, 50, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(75−45, 75, 609)-Net over F₁₂₈ — Digital

Digital (30, 75, 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]

(75−45, 75, 872834)-Net in Base 128 — Upper bound on s

There is no (30, 75, 872835)-net in base 128, because

1 times m-reduction [i] would yield (30, 74, 872835)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 858103 085428 964690 686861 466259 359609 917200 944123 493024 852360 006198 362009 780398 255512 464446 443472 568083 850466 409474 566462 105833 518797 321246 990586 199227 092192 > 128⁷⁴ [i]