MinT - Best Known (68−55, 68, s)-Nets in Base 128

Best Known (68−55, 68, s)-Nets in Base 128

(68−55, 68, 288)-Net over F₁₂₈ — Constructive and digital

Digital (13, 68, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 68, 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]

(68−55, 68, 321)-Net over F₁₂₈ — Digital

Digital (13, 68, 321)-net over F₁₂₈, using

t-expansion [i] based on digital (12, 68, 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]

(68−55, 68, 14561)-Net in Base 128 — Upper bound on s

There is no (13, 68, 14562)-net in base 128, because

1 times m-reduction [i] would yield (13, 67, 14562)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1525 169455 127916 569129 938510 164842 222224 607170 959516 336275 176100 370733 904806 397629 305468 579544 503086 288632 554562 954250 891863 532631 061604 172592 > 128⁶⁷ [i]