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

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

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

Digital (83, 166, 130)-net over F₈, using

trace code for nets [i] based on digital (0, 83, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

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

(83, 166, 225)-net in base 8, using

6 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, 83+83, 266)-Net over F₈ — Digital

Digital (83, 166, 266)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(83, 83+83, 9909)-Net in Base 8 — Upper bound on s

There is no (83, 166, 9910)-net in base 8, because

1 times m-reduction [i] would yield (83, 165, 9910)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102322 906214 624677 531853 491837 383846 627886 217035 151452 888519 185148 671815 122748 824633 329095 489370 109579 868615 463931 417184 609164 374012 347003 411274 893610 > 8¹⁶⁵ [i]