MinT - Best Known (156−61, 156, s)-Nets in Base 8

Best Known (156−61, 156, s)-Nets in Base 8

(156−61, 156, 354)-Net over F₈ — Constructive and digital

Digital (95, 156, 354)-net over F₈, using

t-expansion [i] based on digital (93, 156, 354)-net over F₈, using
- 16 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(156−61, 156, 432)-Net in Base 8 — Constructive

(95, 156, 432)-net in base 8, using

8² times duplication [i] based on (93, 154, 432)-net in base 8, using
- trace code for nets [i] based on (16, 77, 216)-net in base 64, using
  - base change [i] based on digital (5, 66, 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]

(156−61, 156, 737)-Net over F₈ — Digital

Digital (95, 156, 737)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(156−61, 156, 79717)-Net in Base 8 — Upper bound on s

There is no (95, 156, 79718)-net in base 8, because

1 times m-reduction [i] would yield (95, 155, 79718)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 95 288972 859818 582212 580167 079422 141104 245350 270247 170705 212004 338586 541547 587283 477885 573903 546400 872246 419646 733541 174015 847060 994901 333136 > 8¹⁵⁵ [i]