MinT - Best Known (58−45, 58, s)-Nets in Base 128

Best Known (58−45, 58, s)-Nets in Base 128

(58−45, 58, 288)-Net over F₁₂₈ — Constructive and digital

Digital (13, 58, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 58, 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]

(58−45, 58, 321)-Net over F₁₂₈ — Digital

Digital (13, 58, 321)-net over F₁₂₈, using

t-expansion [i] based on digital (12, 58, 321)-net over F₁₂₈, using
- net from sequence [i] based on digital (12, 320)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 12 and N(F) ≥ 321, using
    - a curve from the manYPoints database [i]

(58−45, 58, 20531)-Net in Base 128 — Upper bound on s

There is no (13, 58, 20532)-net in base 128, because

1 times m-reduction [i] would yield (13, 57, 20532)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1 292313 392985 053103 062170 347486 038081 062381 222342 000864 630490 508019 270368 709173 469534 313982 838164 193742 364760 497264 986152 > 128⁵⁷ [i]