MinT - Best Known (84−29, 84, s)-Nets in Base 8

Best Known (84−29, 84, s)-Nets in Base 8

(84−29, 84, 354)-Net over F₈ — Constructive and digital

Digital (55, 84, 354)-net over F₈, using

12 times m-reduction [i] based on digital (55, 96, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 48, 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]

(84−29, 84, 518)-Net in Base 8 — Constructive

(55, 84, 518)-net in base 8, using

base change [i] based on digital (34, 63, 518)-net over F₁₆, using
- 1 times m-reduction [i] based on digital (34, 64, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 32, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(84−29, 84, 840)-Net over F₈ — Digital

Digital (55, 84, 840)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(84−29, 84, 195152)-Net in Base 8 — Upper bound on s

There is no (55, 84, 195153)-net in base 8, because

1 times m-reduction [i] would yield (55, 83, 195153)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 904 653501 067806 821948 107770 124156 690273 699087 500281 960689 934290 902625 959576 > 8⁸³ [i]