MinT - Best Known (91−31, 91, s)-Nets in Base 8

Best Known (91−31, 91, s)-Nets in Base 8

(91−31, 91, 363)-Net over F₈ — Constructive and digital

Digital (60, 91, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (45, 76, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 38, 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]

(91−31, 91, 518)-Net in Base 8 — Constructive

(60, 91, 518)-net in base 8, using

1 times m-reduction [i] based on (60, 92, 518)-net in base 8, using
- base change [i] based on digital (37, 69, 518)-net over F₁₆, using
  - 1 times m-reduction [i] based on digital (37, 70, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 35, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(91−31, 91, 959)-Net over F₈ — Digital

Digital (60, 91, 959)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(91−31, 91, 240542)-Net in Base 8 — Upper bound on s

There is no (60, 91, 240543)-net in base 8, because

1 times m-reduction [i] would yield (60, 90, 240543)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1897 183923 686812 588444 456875 735132 320102 501820 393063 413262 091750 268954 461834 715800 > 8⁹⁰ [i]