MinT - Best Known (116−39, 116, s)-Nets in Base 8

Best Known (116−39, 116, s)-Nets in Base 8

(116−39, 116, 382)-Net over F₈ — Constructive and digital

Digital (77, 116, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 24, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (53, 92, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 46, 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]

(116−39, 116, 576)-Net in Base 8 — Constructive

(77, 116, 576)-net in base 8, using

2 times m-reduction [i] based on (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]

(116−39, 116, 1245)-Net over F₈ — Digital

Digital (77, 116, 1245)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(116−39, 116, 331263)-Net in Base 8 — Upper bound on s

There is no (77, 116, 331264)-net in base 8, because

1 times m-reduction [i] would yield (77, 115, 331264)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 71 675464 813685 055614 020000 309716 959459 510421 526437 084745 543096 015414 150926 357611 979257 924813 317565 203553 > 8¹¹⁵ [i]