MinT - Best Known (77−61, 77, s)-Nets in Base 128

Best Known (77−61, 77, s)-Nets in Base 128

(77−61, 77, 288)-Net over F₁₂₈ — Constructive and digital

Digital (16, 77, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 77, 288)-net over F₁₂₈, using
- net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
    - a function field by SÃ©mirat [i]

(77−61, 77, 386)-Net over F₁₂₈ — Digital

Digital (16, 77, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 77, 386)-net over F₁₂₈, using
- net from sequence [i] based on digital (15, 385)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386, using
    - a curve from the manYPoints database [i]

(77−61, 77, 20650)-Net in Base 128 — Upper bound on s

There is no (16, 77, 20651)-net in base 128, because

1 times m-reduction [i] would yield (16, 76, 20651)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 14059 722760 407604 177955 449810 016681 246866 492897 310372 422507 085821 187915 091023 133903 938764 123272 880812 057821 455876 023070 766057 400835 815337 009948 422340 574772 987960 > 128⁷⁶ [i]