MinT - Best Known (69, 69+36, s)-Nets in Base 8

Best Known (69, 69+36, s)-Nets in Base 8

(69, 69+36, 368)-Net over F₈ — Constructive and digital

Digital (69, 105, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 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 (50, 86, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 43, 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]

(69, 69+36, 520)-Net in Base 8 — Constructive

(69, 105, 520)-net in base 8, using

8¹ times duplication [i] based on (68, 104, 520)-net in base 8, using
- base change [i] based on digital (42, 78, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 39, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(69, 69+36, 1035)-Net over F₈ — Digital

Digital (69, 105, 1035)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+36, 200001)-Net in Base 8 — Upper bound on s

There is no (69, 105, 200002)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 66753 572663 326560 904883 782574 969994 236950 444848 755118 328152 619248 070202 144772 391206 603467 803232 > 8¹⁰⁵ [i]