MinT - Best Known (78, 78+32, s)-Nets in Base 8

Best Known (78, 78+32, s)-Nets in Base 8

(78, 78+32, 484)-Net over F₈ — Constructive and digital

Digital (78, 110, 484)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (16, 32, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 16, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. digital (46, 78, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 39, 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]

(78, 78+32, 576)-Net in Base 8 — Constructive

(78, 110, 576)-net in base 8, using

10 times m-reduction [i] based on (78, 120, 576)-net in base 8, using
- trace code for nets [i] based on (18, 60, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(78, 78+32, 2856)-Net over F₈ — Digital

Digital (78, 110, 2856)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+32, 1571019)-Net in Base 8 — Upper bound on s

There is no (78, 110, 1571020)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2187 256678 958908 871011 597760 619247 780688 290653 796896 131371 751484 703993 878528 836463 126277 352052 578300 > 8¹¹⁰ [i]