MinT - Best Known (99−33, 99, s)-Nets in Base 8

Best Known (99−33, 99, s)-Nets in Base 8

(99−33, 99, 378)-Net over F₈ — Constructive and digital

Digital (66, 99, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 19, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (47, 80, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 40, 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]

(99−33, 99, 576)-Net in Base 8 — Constructive

(66, 99, 576)-net in base 8, using

8¹ times duplication [i] based on (65, 98, 576)-net in base 8, using
- trace code for nets [i] based on (16, 49, 288)-net in base 64, using
  - base change [i] based on digital (9, 42, 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]

(99−33, 99, 1153)-Net over F₈ — Digital

Digital (66, 99, 1153)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(99−33, 99, 330258)-Net in Base 8 — Upper bound on s

There is no (66, 99, 330259)-net in base 8, because

1 times m-reduction [i] would yield (66, 98, 330259)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 31829 478243 116488 008464 661463 298402 774000 080842 109964 765104 187244 633113 905166 722792 967929 > 8⁹⁸ [i]