MinT - Best Known (65, 65+28, s)-Nets in Base 8

Best Known (65, 65+28, s)-Nets in Base 8

(65, 65+28, 399)-Net over F₈ — Constructive and digital

Digital (65, 93, 399)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 23, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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 (42, 70, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 35, 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]

(65, 65+28, 576)-Net in Base 8 — Constructive

(65, 93, 576)-net in base 8, using

5 times m-reduction [i] based on (65, 98, 576)-net in base 8, using
- trace code for nets [i] based on (16, 49, 288)-net in base 64, using
  - base change [i] based on digital (9, 42, 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]

(65, 65+28, 2027)-Net over F₈ — Digital

Digital (65, 93, 2027)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+28, 861892)-Net in Base 8 — Upper bound on s

There is no (65, 93, 861893)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 971339 655421 649319 565626 937784 674030 315109 173352 285655 465748 091223 747411 753625 307168 > 8⁹³ [i]