MinT - Best Known (86, 86+39, s)-Nets in Base 8

Best Known (86, 86+39, s)-Nets in Base 8

(86, 86+39, 419)-Net over F₈ — Constructive and digital

Digital (86, 125, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 33, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [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]

(86, 86+39, 576)-Net in Base 8 — Constructive

(86, 125, 576)-net in base 8, using

9 times m-reduction [i] based on (86, 134, 576)-net in base 8, using
- trace code for nets [i] based on (19, 67, 288)-net in base 64, using
  - 3 times m-reduction [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]

(86, 86+39, 2025)-Net over F₈ — Digital

Digital (86, 125, 2025)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+39, 887078)-Net in Base 8 — Upper bound on s

There is no (86, 125, 887079)-net in base 8, because

1 times m-reduction [i] would yield (86, 124, 887079)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 9619 683209 273609 187539 911051 504545 305919 105363 995437 673716 025757 182395 493502 546539 489385 365945 223299 793833 595256 > 8¹²⁴ [i]