MinT - Best Known (159−54, 159, s)-Nets in Base 8

Best Known (159−54, 159, s)-Nets in Base 8

(159−54, 159, 400)-Net over F₈ — Constructive and digital

Digital (105, 159, 400)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 37, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (68, 122, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 61, 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]

(159−54, 159, 576)-Net in Base 8 — Constructive

(105, 159, 576)-net in base 8, using

9 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]

(159−54, 159, 1533)-Net over F₈ — Digital

Digital (105, 159, 1533)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(159−54, 159, 324700)-Net in Base 8 — Upper bound on s

There is no (105, 159, 324701)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 390239 236376 067164 682611 110696 191605 205733 700349 260436 200872 358521 764315 191981 005818 276739 694093 193554 853929 002505 543888 652307 818579 592050 577476 > 8¹⁵⁹ [i]