MinT - Best Known (84, 84+81, s)-Nets in Base 8

Best Known (84, 84+81, s)-Nets in Base 8

(84, 84+81, 160)-Net over F₈ — Constructive and digital

Digital (84, 165, 160)-net over F₈, using

1 times m-reduction [i] based on digital (84, 166, 160)-net over F₈, using
- trace code for nets [i] based on digital (1, 83, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(84, 84+81, 225)-Net in Base 8 — Constructive

(84, 165, 225)-net in base 8, using

t-expansion [i] based on (83, 165, 225)-net in base 8, using
- 7 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+81, 285)-Net over F₈ — Digital

Digital (84, 165, 285)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(84, 84+81, 11334)-Net in Base 8 — Upper bound on s

There is no (84, 165, 11335)-net in base 8, because

1 times m-reduction [i] would yield (84, 164, 11335)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12814 786810 816215 183456 373926 065657 232640 640423 963779 643183 087564 151443 117778 069397 831868 910347 064861 158009 886375 094597 686424 923921 918481 101631 756344 > 8¹⁶⁴ [i]