MinT - Best Known (64, 64+33, s)-Nets in Base 8

Best Known (64, 64+33, s)-Nets in Base 8

(64, 64+33, 368)-Net over F₈ — Constructive and digital

Digital (64, 97, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 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 (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]

(64, 64+33, 520)-Net in Base 8 — Constructive

(64, 97, 520)-net in base 8, using

8¹ times duplication [i] based on (63, 96, 520)-net in base 8, using
- base change [i] based on digital (39, 72, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 36, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(64, 64+33, 1014)-Net over F₈ — Digital

Digital (64, 97, 1014)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 64+33, 254661)-Net in Base 8 — Upper bound on s

There is no (64, 97, 254662)-net in base 8, because

1 times m-reduction [i] would yield (64, 96, 254662)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 497 327391 114957 102657 620079 227914 330436 160728 241843 332534 531689 679584 111344 947162 659940 > 8⁹⁶ [i]