MinT - Best Known (84, 84+83, s)-Nets in Base 8

Best Known (84, 84+83, s)-Nets in Base 8

(84, 84+83, 130)-Net over F₈ — Constructive and digital

Digital (84, 167, 130)-net over F₈, using

1 times m-reduction [i] based on digital (84, 168, 130)-net over F₈, using
- trace code for nets [i] based on digital (0, 84, 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]

(84, 84+83, 225)-Net in Base 8 — Constructive

(84, 167, 225)-net in base 8, using

t-expansion [i] based on (83, 167, 225)-net in base 8, using
- 5 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]

(84, 84+83, 274)-Net over F₈ — Digital

Digital (84, 167, 274)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(84, 84+83, 10426)-Net in Base 8 — Upper bound on s

There is no (84, 167, 10427)-net in base 8, because

1 times m-reduction [i] would yield (84, 166, 10427)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 818828 859709 452014 298538 929096 034837 440817 210919 749955 686514 160940 653864 198903 786515 031681 416579 043121 754516 571835 323967 557367 491008 786715 638280 780630 > 8¹⁶⁶ [i]