MinT - Best Known (77, 77+41, s)-Nets in Base 8

Best Known (77, 77+41, s)-Nets in Base 8

(77, 77+41, 371)-Net over F₈ — Constructive and digital

Digital (77, 118, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 22, 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 (55, 96, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 48, 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]

(77, 77+41, 576)-Net in Base 8 — Constructive

(77, 118, 576)-net in base 8, using

trace code for nets [i] based on (18, 59, 288)-net in base 64, using
- 4 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]

(77, 77+41, 1059)-Net over F₈ — Digital

Digital (77, 118, 1059)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+41, 227646)-Net in Base 8 — Upper bound on s

There is no (77, 118, 227647)-net in base 8, because

1 times m-reduction [i] would yield (77, 117, 227647)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4587 185246 466056 220092 205141 411862 771850 566507 318645 392559 407183 019493 063137 679789 438371 965952 584975 290681 > 8¹¹⁷ [i]