MinT - Best Known (106−28, 106, s)-Nets in Base 8

Best Known (106−28, 106, s)-Nets in Base 8

(106−28, 106, 562)-Net over F₈ — Constructive and digital

Digital (78, 106, 562)-net over F₈, using

t-expansion [i] based on digital (77, 106, 562)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (20, 34, 208)-net over F₈, using
    - trace code for nets [i] based on digital (3, 17, 104)-net over F₆₄, using
      - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
  2. digital (43, 72, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 36, 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]

(106−28, 106, 674)-Net in Base 8 — Constructive

(78, 106, 674)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (16, 30, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 15, 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]
2. (48, 76, 514)-net in base 8, using
  - base change [i] based on digital (29, 57, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (29, 58, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 29, 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]

(106−28, 106, 5493)-Net over F₈ — Digital

Digital (78, 106, 5493)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−28, 106, 5943474)-Net in Base 8 — Upper bound on s

There is no (78, 106, 5943475)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 533997 307694 972145 638749 597786 957445 863096 241762 627969 259768 918456 682403 937691 041154 615272 781196 > 8¹⁰⁶ [i]