MinT - Best Known (156−63, 156, s)-Nets in Base 8

Best Known (156−63, 156, s)-Nets in Base 8

(156−63, 156, 354)-Net over F₈ — Constructive and digital

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]

(156−63, 156, 384)-Net in Base 8 — Constructive

(93, 156, 384)-net in base 8, using

8² times duplication [i] based on (91, 154, 384)-net in base 8, using
- trace code for nets [i] based on (14, 77, 192)-net in base 64, using
  - base change [i] based on digital (3, 66, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(156−63, 156, 632)-Net over F₈ — Digital

Digital (93, 156, 632)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(156−63, 156, 58108)-Net in Base 8 — Upper bound on s

There is no (93, 156, 58109)-net in base 8, because

1 times m-reduction [i] would yield (93, 155, 58109)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 95 293453 809706 911169 028598 933668 817840 173021 103176 084249 969359 532563 601514 344423 499275 768204 264476 068014 411920 428400 505998 287291 102254 066400 > 8¹⁵⁵ [i]