MinT - Best Known (145−47, 145, s)-Nets in Base 8

Best Known (145−47, 145, s)-Nets in Base 8

(145−47, 145, 419)-Net over F₈ — Constructive and digital

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]

(145−47, 145, 576)-Net in Base 8 — Constructive

(98, 145, 576)-net in base 8, using

t-expansion [i] based on (97, 145, 576)-net in base 8, using
- 9 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]

(145−47, 145, 1829)-Net over F₈ — Digital

Digital (98, 145, 1829)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(145−47, 145, 607402)-Net in Base 8 — Upper bound on s

There is no (98, 145, 607403)-net in base 8, because

1 times m-reduction [i] would yield (98, 144, 607403)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11090 850886 503645 835325 474329 215658 774984 026606 666590 992123 346602 052619 205259 622377 652084 104632 886540 193904 708232 490400 605256 514576 > 8¹⁴⁴ [i]