MinT - Best Known (106−32, 106, s)-Nets in Base 8

Best Known (106−32, 106, s)-Nets in Base 8

(106−32, 106, 402)-Net over F₈ — Constructive and digital

Digital (74, 106, 402)-net over F₈, using

1 times m-reduction [i] based on digital (74, 107, 402)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 27, 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 (47, 80, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 40, 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]

(106−32, 106, 576)-Net in Base 8 — Constructive

(74, 106, 576)-net in base 8, using

t-expansion [i] based on (73, 106, 576)-net in base 8, using
- 6 times m-reduction [i] based on (73, 112, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 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]

(106−32, 106, 2188)-Net over F₈ — Digital

Digital (74, 106, 2188)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−32, 106, 934129)-Net in Base 8 — Upper bound on s

There is no (74, 106, 934130)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 533997 171481 385558 558615 807922 467811 455951 222108 411672 790890 364437 372451 687295 922908 163609 134467 > 8¹⁰⁶ [i]