MinT - Best Known (69, 69+97, s)-Nets in Base 8

Best Known (69, 69+97, s)-Nets in Base 8

(69, 69+97, 99)-Net over F₈ — Constructive and digital

Digital (69, 166, 99)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 55, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (14, 111, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(69, 69+97, 144)-Net over F₈ — Digital

Digital (69, 166, 144)-net over F₈, using

t-expansion [i] based on digital (45, 166, 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+97, 156)-Net in Base 8

(69, 166, 156)-net in base 8, using

2 times m-reduction [i] based on (69, 168, 156)-net in base 8, using
- base change [i] based on digital (27, 126, 156)-net over F₁₆, using
  - net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
      - a curve from the manYPoints database [i]

(69, 69+97, 3374)-Net in Base 8 — Upper bound on s

There is no (69, 166, 3375)-net in base 8, because

1 times m-reduction [i] would yield (69, 165, 3375)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 103300 739594 121027 228612 812008 676646 464358 757742 117891 033015 312908 679465 693816 851702 959530 205438 167353 877223 911694 762501 246699 215653 150633 632526 582896 > 8¹⁶⁵ [i]