MinT - Best Known (122−47, 122, s)-Nets in Base 8

Best Known (122−47, 122, s)-Nets in Base 8

(122−47, 122, 354)-Net over F₈ — Constructive and digital

Digital (75, 122, 354)-net over F₈, using

14 times m-reduction [i] based on digital (75, 136, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 68, 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]

(122−47, 122, 432)-Net in Base 8 — Constructive

(75, 122, 432)-net in base 8, using

trace code for nets [i] based on (14, 61, 216)-net in base 64, using
- 2 times m-reduction [i] based on (14, 63, 216)-net in base 64, using
  - base change [i] based on digital (5, 54, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(122−47, 122, 651)-Net over F₈ — Digital

Digital (75, 122, 651)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(122−47, 122, 75912)-Net in Base 8 — Upper bound on s

There is no (75, 122, 75913)-net in base 8, because

1 times m-reduction [i] would yield (75, 121, 75913)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 18 789339 005753 926108 973544 136031 654725 883977 456823 751476 799534 767367 500030 052190 470744 598391 148806 183983 267904 > 8¹²¹ [i]