MinT - Best Known (56, 56+23, s)-Nets in Base 8

Best Known (56, 56+23, s)-Nets in Base 8

(56, 56+23, 389)-Net over F₈ — Constructive and digital

Digital (56, 79, 389)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 19, 35)-net over F₈, using
  - net from sequence [i] based on digital (8, 34)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 7, N(F)Â =Â 34, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using a function field by SÃ©mirat [i]
2. digital (37, 60, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 30, 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]

(56, 56+23, 576)-Net in Base 8 — Constructive

(56, 79, 576)-net in base 8, using

3 times m-reduction [i] based on (56, 82, 576)-net in base 8, using
- trace code for nets [i] based on (15, 41, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
    - base change [i] based on digital (9, 36, 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]

(56, 56+23, 2274)-Net over F₈ — Digital

Digital (56, 79, 2274)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+23, 1776820)-Net in Base 8 — Upper bound on s

There is no (56, 79, 1776821)-net in base 8, because

1 times m-reduction [i] would yield (56, 78, 1776821)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27607 104114 437162 665340 951436 533581 384875 505656 338145 839996 496694 754008 > 8⁷⁸ [i]