MinT - Best Known (69, 69+101, s)-Nets in Base 8

Best Known (69, 69+101, s)-Nets in Base 8

(69, 69+101, 98)-Net over F₈ — Constructive and digital

Digital (69, 170, 98)-net over F₈, using

t-expansion [i] based on digital (37, 170, 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]

(69, 69+101, 144)-Net over F₈ — Digital

Digital (69, 170, 144)-net over F₈, using

t-expansion [i] based on digital (45, 170, 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]

(69, 69+101, 150)-Net in Base 8

(69, 170, 150)-net in base 8, using

2 times m-reduction [i] based on (69, 172, 150)-net in base 8, using
- base change [i] based on digital (26, 129, 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]

(69, 69+101, 3109)-Net in Base 8 — Upper bound on s

There is no (69, 170, 3110)-net in base 8, because

1 times m-reduction [i] would yield (69, 169, 3110)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 425 599873 131575 455512 845970 399484 699002 563806 446575 811689 287651 092570 167427 221451 897058 306866 639828 700517 474517 077265 655826 421281 584670 039934 844200 218496 > 8¹⁶⁹ [i]