MinT - Best Known (99, 99+53, s)-Nets in Base 8

Best Known (99, 99+53, s)-Nets in Base 8

(99, 99+53, 382)-Net over F₈ — Constructive and digital

Digital (99, 152, 382)-net over F₈, using

8¹ times duplication [i] based on digital (98, 151, 382)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 31, 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 (67, 120, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 60, 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+53, 576)-Net in Base 8 — Constructive

(99, 152, 576)-net in base 8, using

t-expansion [i] based on (97, 152, 576)-net in base 8, using
- 2 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+53, 1287)-Net over F₈ — Digital

Digital (99, 152, 1287)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 99+53, 264872)-Net in Base 8 — Upper bound on s

There is no (99, 152, 264873)-net in base 8, because

1 times m-reduction [i] would yield (99, 151, 264873)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 23259 541730 766254 592386 771704 576892 670360 478808 460521 101848 263915 648954 166810 529200 026671 681225 252074 416008 972330 114665 646232 704760 455792 > 8¹⁵¹ [i]