MinT - Best Known (86, 86+45, s)-Nets in Base 8

Best Known (86, 86+45, s)-Nets in Base 8

(86, 86+45, 382)-Net over F₈ — Constructive and digital

Digital (86, 131, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 27, 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 (59, 104, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 52, 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]

(86, 86+45, 576)-Net in Base 8 — Constructive

(86, 131, 576)-net in base 8, using

3 times m-reduction [i] based on (86, 134, 576)-net in base 8, using
- trace code for nets [i] based on (19, 67, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(86, 86+45, 1226)-Net over F₈ — Digital

Digital (86, 131, 1226)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+45, 280656)-Net in Base 8 — Upper bound on s

There is no (86, 131, 280657)-net in base 8, because

1 times m-reduction [i] would yield (86, 130, 280657)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2521 877848 129795 089904 461699 648230 407204 056113 348537 751679 967830 696113 331214 731312 205784 545203 852518 479679 640345 143712 > 8¹³⁰ [i]