MinT - Best Known (69, 69+37, s)-Nets in Base 8

Best Known (69, 69+37, s)-Nets in Base 8

(69, 69+37, 363)-Net over F₈ — Constructive and digital

Digital (69, 106, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 18, 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 (51, 88, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 44, 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+37, 518)-Net in Base 8 — Constructive

(69, 106, 518)-net in base 8, using

8² times duplication [i] based on (67, 104, 518)-net in base 8, using
- base change [i] based on digital (41, 78, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 39, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(69, 69+37, 948)-Net over F₈ — Digital

Digital (69, 106, 948)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+37, 200001)-Net in Base 8 — Upper bound on s

There is no (69, 106, 200002)-net in base 8, because

1 times m-reduction [i] would yield (69, 105, 200002)-net in base 8, but
- 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]