MinT - Best Known (92−31, 92, s)-Nets in Base 8

Best Known (92−31, 92, s)-Nets in Base 8

(92−31, 92, 368)-Net over F₈ — Constructive and digital

Digital (61, 92, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [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]

(92−31, 92, 520)-Net in Base 8 — Constructive

(61, 92, 520)-net in base 8, using

base change [i] based on digital (38, 69, 520)-net over F₁₆, using
- 1 times m-reduction [i] based on digital (38, 70, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 35, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(92−31, 92, 1027)-Net over F₈ — Digital

Digital (61, 92, 1027)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(92−31, 92, 276312)-Net in Base 8 — Upper bound on s

There is no (61, 92, 276313)-net in base 8, because

1 times m-reduction [i] would yield (61, 91, 276313)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 15177 712050 309748 020358 606782 213601 715387 906938 002335 047002 477562 859905 705922 661776 > 8⁹¹ [i]