MinT - Best Known (130−39, 130, s)-Nets in Base 8

Best Known (130−39, 130, s)-Nets in Base 8

(130−39, 130, 484)-Net over F₈ — Constructive and digital

Digital (91, 130, 484)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (19, 38, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 19, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [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]

(130−39, 130, 576)-Net in Base 8 — Constructive

(91, 130, 576)-net in base 8, using

t-expansion [i] based on (89, 130, 576)-net in base 8, using
- 10 times m-reduction [i] based on (89, 140, 576)-net in base 8, using
  - trace code for nets [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]

(130−39, 130, 2657)-Net over F₈ — Digital

Digital (91, 130, 2657)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(130−39, 130, 1533274)-Net in Base 8 — Upper bound on s

There is no (91, 130, 1533275)-net in base 8, because

1 times m-reduction [i] would yield (91, 129, 1533275)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 315 218877 465675 730748 805905 650546 887083 237539 209960 491487 011678 312910 190727 446909 825620 427547 467080 613412 763423 604416 > 8¹²⁹ [i]