MinT - Best Known (81, 81+43, s)-Nets in Base 8

Best Known (81, 81+43, s)-Nets in Base 8

(81, 81+43, 378)-Net over F₈ — Constructive and digital

Digital (81, 124, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (57, 100, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 50, 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]

(81, 81+43, 576)-Net in Base 8 — Constructive

(81, 124, 576)-net in base 8, using

2 times m-reduction [i] based on (81, 126, 576)-net in base 8, using
- trace code for nets [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]

(81, 81+43, 1115)-Net over F₈ — Digital

Digital (81, 124, 1115)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+43, 241489)-Net in Base 8 — Upper bound on s

There is no (81, 124, 241490)-net in base 8, because

1 times m-reduction [i] would yield (81, 123, 241490)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1202 508211 801004 501585 735747 015581 450796 671645 395184 862032 436546 751768 223016 179006 555660 932907 158828 747747 019664 > 8¹²³ [i]