MinT - Best Known (106−33, 106, s)-Nets in Base 8

Best Known (106−33, 106, s)-Nets in Base 8

(106−33, 106, 400)-Net over F₈ — Constructive and digital

Digital (73, 106, 400)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 26, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [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]

(106−33, 106, 576)-Net in Base 8 — Constructive

(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]

(106−33, 106, 1808)-Net over F₈ — Digital

Digital (73, 106, 1808)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−33, 106, 820282)-Net in Base 8 — Upper bound on s

There is no (73, 106, 820283)-net in base 8, because

1 times m-reduction [i] would yield (73, 105, 820283)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 66750 007511 011642 794575 130792 616411 457686 447653 442723 273060 594994 523656 915517 227301 730963 400577 > 8¹⁰⁵ [i]