MinT - Best Known (84, 84+38, s)-Nets in Base 8

Best Known (84, 84+38, s)-Nets in Base 8

(84, 84+38, 402)-Net over F₈ — Constructive and digital

Digital (84, 122, 402)-net over F₈, using

t-expansion [i] based on digital (83, 122, 402)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 30, 48)-net over F₈, using
    - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
        a function field by SÃ©mirat [i]
  2. 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, 84+38, 576)-Net in Base 8 — Constructive

(84, 122, 576)-net in base 8, using

8 times m-reduction [i] based on (84, 130, 576)-net in base 8, using
- trace code for nets [i] based on (19, 65, 288)-net in base 64, using
  - 5 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(84, 84+38, 2007)-Net over F₈ — Digital

Digital (84, 122, 2007)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(84, 84+38, 712686)-Net in Base 8 — Upper bound on s

There is no (84, 122, 712687)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 150 307852 375455 536477 794322 903672 296976 293703 706389 665353 475345 097617 236845 676134 179968 774247 564288 979459 007752 > 8¹²² [i]