MinT - Best Known (83, 168, s)-Nets in Base 8

Best Known (83, 168, s)-Nets in Base 8

(83, 168, 130)-Net over F₈ — Constructive and digital

Digital (83, 168, 130)-net over F₈, using

t-expansion [i] based on digital (76, 168, 130)-net over F₈, using
- 4 times m-reduction [i] based on digital (76, 172, 130)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 62, 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]
    2. digital (14, 110, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(83, 168, 225)-Net in Base 8 — Constructive

(83, 168, 225)-net in base 8, using

4 times m-reduction [i] based on (83, 172, 225)-net in base 8, using
- base change [i] based on digital (40, 129, 225)-net over F₁₆, using
  - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
      - a function field by GarcÃa and Quoos [i]

(83, 168, 256)-Net over F₈ — Digital

Digital (83, 168, 256)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(83, 168, 9168)-Net in Base 8 — Upper bound on s

There is no (83, 168, 9169)-net in base 8, because

1 times m-reduction [i] would yield (83, 167, 9169)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 554778 165895 787418 701789 014146 916337 371592 174012 435542 746579 057680 706815 400939 040716 918163 540268 709073 600877 434924 786079 462129 929371 788264 163140 076240 > 8¹⁶⁷ [i]