MinT - Best Known (74, 74+31, s)-Nets in Base 8

Best Known (74, 74+31, s)-Nets in Base 8

(74, 74+31, 419)-Net over F₈ — Constructive and digital

Digital (74, 105, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 29, 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 (45, 76, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 38, 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]

(74, 74+31, 576)-Net in Base 8 — Constructive

(74, 105, 576)-net in base 8, using

t-expansion [i] based on (73, 105, 576)-net in base 8, using
- 7 times m-reduction [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]

(74, 74+31, 2507)-Net over F₈ — Digital

Digital (74, 105, 2507)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(74, 74+31, 1675290)-Net in Base 8 — Upper bound on s

There is no (74, 105, 1675291)-net in base 8, because

1 times m-reduction [i] would yield (74, 104, 1675291)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8343 732495 777401 871238 198917 561505 146299 505859 566361 609443 381169 034441 551147 426333 465918 802960 > 8¹⁰⁴ [i]