MinT - Best Known (99, 99+47, s)-Nets in Base 8

Best Known (99, 99+47, s)-Nets in Base 8

(99, 99+47, 419)-Net over F₈ — Constructive and digital

Digital (99, 146, 419)-net over F₈, using

8¹ times duplication [i] based on digital (98, 145, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 37, 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 (61, 108, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 54, 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]

(99, 99+47, 576)-Net in Base 8 — Constructive

(99, 146, 576)-net in base 8, using

t-expansion [i] based on (97, 146, 576)-net in base 8, using
- 8 times m-reduction [i] based on (97, 154, 576)-net in base 8, using
  - trace code for nets [i] based on (20, 77, 288)-net in base 64, using
    - base change [i] based on digital (9, 66, 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]

(99, 99+47, 1913)-Net over F₈ — Digital

Digital (99, 146, 1913)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 99+47, 664878)-Net in Base 8 — Upper bound on s

There is no (99, 146, 664879)-net in base 8, because

1 times m-reduction [i] would yield (99, 145, 664879)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 88726 894114 236128 660066 146572 680415 145241 119548 472756 582705 587953 819599 404067 594600 557423 374929 672031 450665 496983 608679 422227 645280 > 8¹⁴⁵ [i]