MinT - Best Known (99, 99+45, s)-Nets in Base 8

Best Known (99, 99+45, s)-Nets in Base 8

(99, 99+45, 419)-Net over F₈ — Constructive and digital

Digital (99, 144, 419)-net over F₈, using

t-expansion [i] based on digital (98, 144, 419)-net over F₈, using
- 1 times m-reduction [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+45, 576)-Net in Base 8 — Constructive

(99, 144, 576)-net in base 8, using

t-expansion [i] based on (97, 144, 576)-net in base 8, using
- 10 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+45, 2247)-Net over F₈ — Digital

Digital (99, 144, 2247)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 99+45, 959033)-Net in Base 8 — Upper bound on s

There is no (99, 144, 959034)-net in base 8, because

1 times m-reduction [i] would yield (99, 143, 959034)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1386 354502 164882 688994 752967 568873 354307 926785 630515 636932 981703 484248 251313 843304 870392 665762 590577 142509 185781 028076 317768 864244 > 8¹⁴³ [i]