MinT - Best Known (113−39, 113, s)-Nets in Base 8

Best Known (113−39, 113, s)-Nets in Base 8

(113−39, 113, 371)-Net over F₈ — Constructive and digital

Digital (74, 113, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 21, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [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]

(113−39, 113, 576)-Net in Base 8 — Constructive

(74, 113, 576)-net in base 8, using

8¹ times duplication [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]

(113−39, 113, 1059)-Net over F₈ — Digital

Digital (74, 113, 1059)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(113−39, 113, 238546)-Net in Base 8 — Upper bound on s

There is no (74, 113, 238547)-net in base 8, because

1 times m-reduction [i] would yield (74, 112, 238547)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 139984 050860 459496 519862 053008 906209 741292 951282 285008 362594 981960 926213 680110 093585 866743 767201 151032 > 8¹¹² [i]