MinT - Best Known (84−31, 84, s)-Nets in Base 8

Best Known (84−31, 84, s)-Nets in Base 8

(84−31, 84, 354)-Net over F₈ — Constructive and digital

Digital (53, 84, 354)-net over F₈, using

8 times m-reduction [i] based on digital (53, 92, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 46, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(84−31, 84, 514)-Net in Base 8 — Constructive

(53, 84, 514)-net in base 8, using

base change [i] based on digital (32, 63, 514)-net over F₁₆, using
- 1 times m-reduction [i] based on digital (32, 64, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 32, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(84−31, 84, 596)-Net over F₈ — Digital

Digital (53, 84, 596)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(84−31, 84, 91142)-Net in Base 8 — Upper bound on s

There is no (53, 84, 91143)-net in base 8, because

1 times m-reduction [i] would yield (53, 83, 91143)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 904 636714 818118 480229 096511 773164 172066 602486 095138 038612 844185 465963 427840 > 8⁸³ [i]