MinT - Best Known (160−53, 160, s)-Nets in Base 8

Best Known (160−53, 160, s)-Nets in Base 8

(160−53, 160, 419)-Net over F₈ — Constructive and digital

Digital (107, 160, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 40, 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 (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]

(160−53, 160, 576)-Net in Base 8 — Constructive

(107, 160, 576)-net in base 8, using

t-expansion [i] based on (105, 160, 576)-net in base 8, using
- 8 times m-reduction [i] based on (105, 168, 576)-net in base 8, using
  - trace code for nets [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(160−53, 160, 1762)-Net over F₈ — Digital

Digital (107, 160, 1762)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(160−53, 160, 502254)-Net in Base 8 — Upper bound on s

There is no (107, 160, 502255)-net in base 8, because

1 times m-reduction [i] would yield (107, 159, 502255)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 390234 375436 509398 531728 121462 901150 131118 791431 975776 751351 364650 729084 182613 390646 086198 044547 054899 745510 579189 240798 282030 499668 436951 343478 > 8¹⁵⁹ [i]