MinT - Best Known (68−53, 68, s)-Nets in Base 128

Best Known (68−53, 68, s)-Nets in Base 128

(68−53, 68, 288)-Net over F₁₂₈ — Constructive and digital

Digital (15, 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]

(68−53, 68, 386)-Net over F₁₂₈ — Digital

Digital (15, 68, 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]

(68−53, 68, 22354)-Net in Base 128 — Upper bound on s

There is no (15, 68, 22355)-net in base 128, because

1 times m-reduction [i] would yield (15, 67, 22355)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1525 576432 078899 910014 208305 778083 017951 823021 065484 480193 582097 526590 546431 341500 939239 000761 408590 076500 123266 800863 387137 302372 859291 640936 > 128⁶⁷ [i]