MinT - Best Known (45, 45+23, s)-Nets in Base 8

Best Known (45, 45+23, s)-Nets in Base 8

(45, 45+23, 354)-Net over F₈ — Constructive and digital

Digital (45, 68, 354)-net over F₈, using

8 times m-reduction [i] based on 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]

(45, 45+23, 518)-Net in Base 8 — Constructive

(45, 68, 518)-net in base 8, using

base change [i] based on digital (28, 51, 518)-net over F₁₆, using
- 1 times m-reduction [i] based on digital (28, 52, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 26, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(45, 45+23, 811)-Net over F₈ — Digital

Digital (45, 68, 811)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+23, 222096)-Net in Base 8 — Upper bound on s

There is no (45, 68, 222097)-net in base 8, because

1 times m-reduction [i] would yield (45, 67, 222097)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 213909 796964 240622 289174 566779 665517 357270 180766 182348 086960 > 8⁶⁷ [i]