MinT - Best Known (66, 66+93, s)-Nets in Base 8

Best Known (66, 66+93, s)-Nets in Base 8

(66, 66+93, 98)-Net over F₈ — Constructive and digital

Digital (66, 159, 98)-net over F₈, using

t-expansion [i] based on digital (37, 159, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(66, 66+93, 144)-Net over F₈ — Digital

Digital (66, 159, 144)-net over F₈, using

t-expansion [i] based on digital (45, 159, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(66, 66+93, 150)-Net in Base 8

(66, 159, 150)-net in base 8, using

1 times m-reduction [i] based on (66, 160, 150)-net in base 8, using
- base change [i] based on digital (26, 120, 150)-net over F₁₆, using
  - net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
      - a curve from the manYPoints database [i]

(66, 66+93, 3222)-Net in Base 8 — Upper bound on s

There is no (66, 159, 3223)-net in base 8, because

1 times m-reduction [i] would yield (66, 158, 3223)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 49296 921310 689447 438815 420544 904012 237173 504988 731700 990963 334680 875665 566034 142037 143549 249795 735107 810568 171937 810280 111632 848351 457948 384408 > 8¹⁵⁸ [i]